1300 Math Formulas(MATHS 4 ALL)

CHAPTER 9. INTEGRAL CALCULUS

fñò = ∫∫ óOµ(ñI óIò)Çp K=
p

=
1156. jçãÉåíë=çÑ=fåÉêíá~=~Äçìí=íÜÉ=ñ-~ñáëI=ó-~ñáëI=~åÇ=ò-~ñáë=

fñ = ∫∫(óO + òO )µ(ñIóIò)Çp I==
p

fó = ∫∫(ñO + òO )µ(ñIóIò)Çp I==
p

fò = ∫∫(ñO + )óO µ(ñIóIò)Çp K==
p

=
1157. sçäìãÉ=çÑ=~=pçäáÇ=_çìåÇÉÇ=Äó=~=`äçëÉÇ=pìêÑ~ÅÉ=

s = N ∫∫ ñÇóÇò + óÇñÇò + òÇñÇó ==
P
p

=

1158. dcr ê=~dîáãí~∫ípá∫çµå(~ñäI=cóçIòê)ÅêÉrêP= Çp I=

ïÜÉêÉ=ã=áë=~=ã~ëë=~í=~=éçáåí= ñMIóMIòM =çìíëáÇÉ=íÜÉ=ëìêÑ~ÅÉI==
rê = ñ − ñM I ó − óM Iò − òM I==

µ(ñI óIò) =áë=íÜÉ=ÇÉåëáíó=ÑìåÅíáçåI==

~åÇ=d=áë=Öê~îáí~íáçå~ä=Åçåëí~åíK=

=

1159. mcrê=Éë∫ë∫ìéê(Érê=c)ÇçprêÅI=É==
p
=rêéK=(rê ) ==~Åíë= = çå== íÜÉ= = ëìêÑ~ÅÉ==p==ÖáîÉå== Äó=
ïÜÉêÉ=íÜÉ=éêÉëëìêÉ=
íÜÉ=éçëáíáçå=îÉÅíçê=

=

1160. cΦäì=áÇ∫=∫cîräì(rêñ)=⋅EÇ~ÅprêIç==ëë=íÜÉ=ëìêÑ~ÅÉ=pF=
p

292

CHAPTER 9. INTEGRAL CALCULUS

ïÜÉêÉ= îr(rê)=áë=íÜÉ=ÑäìáÇ=îÉäçÅáíóK=

==

1161. jΦ ~=ëë∫=∫cρäìîrñ(rê=E)~⋅ÅÇêprçIë=ë==íÜÉ=ëìêÑ~ÅÉ=pF=
p
cr = ρîr =áë=íÜÉ=îÉÅíçê=ÑáÉäÇI= ρ =áë=íÜÉ=ÑäìáÇ=ÇÉåëáíóK=
ïÜÉêÉ=

=
1162. pìêÑ~ÅÉ=`Ü~êÖÉ=

n = ∫∫σ(ñIó)Çp I==
p

ïÜÉêÉ= σ(ñI ó)=áë=íÜÉ=ëìêÑ~ÅÉ=ÅÜ~êÖÉ=ÇÉåëáíóK=

=
1163. d~ìëë∞=i~ï=

qÜÉ=ÉäÉÅíêáÅ=Ñäìñ=íÜêçìÖÜ=~åó=ÅäçëÉÇ=ëìêÑ~ÅÉ=áë=éêçéçêíáçå~ä=

íç=íÜÉ=ÅÜ~êÖÉ=n=ÉåÅäçëÉÇ=Äó=íÜÉ=ëìêÑ~ÅÉ=
∫∫br ⋅Çpr
Φ = = n I==
p εM

ïÜÉêÉ==

Φbr ==ááëë==ííÜÜÉÉ==ãÉä~ÉÖÅíåêááíÅì=ÑÇäìÉ=ñçIÑ===íÜÉ=ÉäÉÅíêáÅ=ÑáÉäÇ=ëíêÉåÖíÜI=

εM = UIUR × NM −NO c =áë=éÉêãáííáîáíó=çÑ=ÑêÉÉ=ëé~ÅÉK==
ã

=

=

293

Chapter 10

Differential Equations

=
=
=
=
cìåÅíáçåë=çÑ=çåÉ=î~êá~ÄäÉW=óI=éI=èI=ìI=ÖI=ÜI=dI=eI=êI=ò==
^êÖìãÉåíë=EáåÇÉéÉåÇÉåí=î~êá~ÄäÉëFW=ñI=ó=

cìåÅíáçåë=çÑ=íïç=î~êá~ÄäÉëW= Ñ (ñIó) I= j(ñIó)I= k(ñIó) =

cáêëí=çêÇÉê=ÇÉêáî~íáîÉW= ó′ I= ì′ I= ó& I= Çó I=£=
Çí

pÉÅçåÇ=çêÇÉê=ÇÉêáî~íáîÉëW= ó′′ I= &ó& I= ÇOf I=£=
ÇíO

m~êíá~ä=ÇÉêáî~íáîÉëW= ∂ì I= ∂Oì I=£=
∂í ∂ñ O

k~íìê~ä=åìãÄÉêW=å=

m~êíáÅìä~ê=ëçäìíáçåëW= óN I= óé =

oÉ~ä=åìãÄÉêëW=âI=íI=`I= `NI= `O I=éI=èI= α I= β =
oççíë=çÑ=íÜÉ=ÅÜ~ê~ÅíÉêáëíáÅ=Éèì~íáçåëW= λNI= λO =
qáãÉW=í=
qÉãéÉê~íìêÉW=qI=p=

mçéìä~íáçå=ÑìåÅíáçåW= m(í) ==

j~ëë=çÑ=~å=çÄàÉÅíW=ã=
píáÑÑåÉëë=çÑ=~=ëéêáåÖW=â=
aáëéä~ÅÉãÉåí=çÑ=íÜÉ=ã~ëë=Ñêçã=ÉèìáäáÄêáìãW=ó=

^ãéäáíìÇÉ=çÑ=íÜÉ=Çáëéä~ÅÉãÉåíW=^=
cêÉèìÉåÅóW= ω=
a~ãéáåÖ=ÅçÉÑÑáÅáÉåíW= γ =

mÜ~ëÉ=~åÖäÉ=çÑ=íÜÉ=Çáëéä~ÅÉãÉåíW= δ =
^åÖìä~ê=Çáëéä~ÅÉãÉåíW= θ =
mÉåÇìäìã=äÉåÖíÜW=i=

294

CHAPTER 10. DIFFERENTIAL EQUATIONS

^ÅÅÉäÉê~íáçå=çÑ=Öê~îáíóW=Ö=
`ìêêÉåíW=f=

oÉëáëí~åÅÉW=o=
fåÇìÅí~åÅÉW=i=

`~é~Åáí~åÅÉW=`=
=
=

10.1 First Order Ordinary Differential
Equations

=
1164. iáåÉ~ê=bèì~íáçåë=

Çó + é(ñ)ó = è(ñ)K==

Çñ
=

qÜÉ=ÖÉåÉê~ä=ëçäìíáçå=áë==

ó = ∫ ì(ñ )è(ñ )Çñ + ` I==
ì(ñ)

ïÜÉêÉ==

( )ì(ñ) = Éñé ∫ é(ñ)Çñ K=

=
1165. pÉé~ê~ÄäÉ=bèì~íáçåë=

Çó = Ñ (ñIó) = Ö(ñ)Ü(ó)=

Çñ
=

qÜÉ=ÖÉåÉê~ä=ëçäìíáçå=áë=ÖáîÉå=Äó=

∫ Çó ) = ∫ Ö(ñ )Çñ + ` I==

Ü(ó

çê=

e(ó) = d(ñ)+ ` K=

=
=
=

295

CHAPTER 10. DIFFERENTIAL EQUATIONS

1166. eçãçÖÉåÉçìë=bèì~íáçåë=

qÜÉ= ÇáÑÑÉêÉåíá~ä= Éèì~íáçå= Çó = Ñ (ñI ó) = áë= ÜçãçÖÉåÉçìëI= áÑ=

Çñ

íÜÉ=ÑìåÅíáçå= Ñ (ñI ó) =áë=ÜçãçÖÉåÉçìëI=íÜ~í=áë==

Ñ (íñIíó) = Ñ (ñIó) K==

=

qÜÉ=ëìÄëíáíìíáçå= ò = ó =EíÜÉå= ó = òñ F=äÉ~Çë=íç=íÜÉ=ëÉé~ê~ÄäÉ=
ñ

Éèì~íáçå=

ñ Çò + ò = Ñ (NIò)K==

Çñ

=
1167. _Éêåçìääá=bèì~íáçå=

Çó + é(ñ)ó = è(ñ)óå K==

Çñ
=

qÜÉ=ëìÄëíáíìíáçå= ò = óN−å =äÉ~Çë=íç=íÜÉ=äáåÉ~ê=Éèì~íáçå==

Çò + (N − å)é(ñ ) ò = (N − å )è(ñ ) K==
Çñ

=

1168. oáÅÅ~íá=bèì~íáçå=

Çó = é(ñ)+ è(ñ) ó + ê(ñ) óO =

Çñ
=
fÑ=~=é~êíáÅìä~ê=ëçäìíáçå= óN =áë=âåçïåI=íÜÉå=íÜÉ=ÖÉåÉê~ä=ëçäì-

íáçå=Å~å=ÄÉ=çÄí~áåÉÇ=ïáíÜ=íÜÉ=ÜÉäé=çÑ=ëìÄëíáíìíáçå=

ò = N I=ïÜáÅÜ=äÉ~Çë=íç=íÜÉ=Ñáêëí=çêÇÉê=äáåÉ~ê=Éèì~íáçå==
ó − óN

Çò = −[è(ñ ) + OóNê(ñ )] ò − ê(ñ ) K==
Çñ

=

=

=

296

CHAPTER 10. DIFFERENTIAL EQUATIONS

1169. bñ~Åí=~åÇ=kçåÉñ~Åí=bèì~íáçåë=
qÜÉ=Éèì~íáçå==

j(ñIó)Çñ + k(ñIó)Çó = M ==

áë=Å~ääÉÇ=Éñ~Åí=áÑ==
∂j = ∂k I==
∂ó ∂ñ
~åÇ=åçåÉñ~Åí=çíÜÉêïáëÉK=
=
qÜÉ=ÖÉåÉê~ä=ëçäìíáçå=áë==

∫ j(ñIó)Çñ + ∫ k(ñIó)Çó = ` K==

=
1170. o~Çáç~ÅíáîÉ=aÉÅ~ó=

Çó = −âó I==
Çí

ïÜÉêÉ= ó(í) =áë=íÜÉ=~ãçìåí=çÑ=ê~Çáç~ÅíáîÉ=ÉäÉãÉåí=~í=íáãÉ=íI=â=

áë=íÜÉ=ê~íÉ=çÑ=ÇÉÅ~óK==
=
qÜÉ=ëçäìíáçå=áë==

ó(í) = óMÉ−âí I=ïÜÉêÉ= óM = ó(M) =áë=íÜÉ=áåáíá~ä=~ãçìåíK=

=
1171. kÉïíçå∞ë=i~ï=çÑ=`ççäáåÖ=

Çq = −â(q − p)I==

Çí

ïÜÉêÉ= q(í) =áë=íÜÉ=íÉãéÉê~íìêÉ=çÑ=~å=çÄàÉÅí=~í=íáãÉ=íI=p=áë=íÜÉ=

íÉãéÉê~íìêÉ= çÑ= íÜÉ= ëìêêçìåÇáåÖ= ÉåîáêçåãÉåíI= â= áë= ~= éçëá-
íáîÉ=Åçåëí~åíK==
=
qÜÉ=ëçäìíáçå=áë=

q(í) = p + (qM − p) É−âí I==
ïÜÉêÉ= qM = q(M) = áë= íÜÉ= áåáíá~ä= íÉãéÉê~íìêÉ= çÑ= íÜÉ= çÄàÉÅí= ~í=

íáãÉ= í = M K==
=
=

297

CHAPTER 10. DIFFERENTIAL EQUATIONS

1172. mçéìä~íáçå=aóå~ãáÅë=EiçÖáëíáÅ=jçÇÉäF=

Çm = âmN − m  I==
Çí  j 

ïÜÉêÉ= m(í) =áë=éçéìä~íáçå=~í=íáãÉ=íI=â=áë=~=éçëáíáîÉ=Åçåëí~åíI=

j=áë=~=äáãáíáåÖ=ëáòÉ=Ñçê=íÜÉ=éçéìä~íáçåK==

=

qÜÉ=ëçäìíáçå=çÑ=íÜÉ=ÇáÑÑÉêÉåíá~ä=Éèì~íáçå=áë==

m(í) = mM jmM I=ïÜÉêÉ= mM = m(M)=áë=íÜÉ=áåáíá~ä=éçéì-

+ (j − mM )É−âí

ä~íáçå=~í=íáãÉ= í = M K===

=

=

=

10.2 Second Order Ordinary Differential

Equations

=
1173. eçãçÖÉåÉçìë=iáåÉ~ê=bèì~íáçåë=ïáíÜ=`çåëí~åí=`çÉÑÑáÅáÉåíë==

ó′′ + éó′ + èó = M K==

qÜÉ=ÅÜ~ê~ÅíÉêáëíáÅ=Éèì~íáçå=áë==

λO + éλ + è = M K==

=

fÑ= λN= ~åÇ= λO = ~êÉ= ÇáëíáåÅí= êÉ~ä= êççíë= çÑ= íÜÉ= ÅÜ~ê~ÅíÉêáëíáÅ=
Éèì~íáçåI=íÜÉå=íÜÉ=ÖÉåÉê~ä=ëçäìíáçå=áë==

ó = `NÉλNñ + `OÉλOñ I=ïÜÉêÉ==
`N=~åÇ= `O =~êÉ=áåíÉÖê~íáçå=Åçåëí~åíëK==
==

fÑ= λN = λO = − é I=íÜÉå=íÜÉ=ÖÉåÉê~ä=ëçäìíáçå=áë==
O

ó = (`N + `Oñ )É− é ñ K==
O

=

fÑ= λN=~åÇ= λO =~êÉ=ÅçãéäÉñ=åìãÄÉêëW=

298

CHAPTER 10. DIFFERENTIAL EQUATIONS

λN = α + βá I= λO = α − βá I=ïÜÉêÉ==

α = − é I= β = Qè − éO I==
OO

íÜÉå=íÜÉ=ÖÉåÉê~ä=ëçäìíáçå=áë==

ó = Éαñ (`N Åçëβñ + `O ëáåβñ) K==

=
1174. fåÜçãçÖÉåÉçìë=iáåÉ~ê=bèì~íáçåë=ïáíÜ=`çåëí~åí=======================

`çÉÑÑáÅáÉåíë==

ó′′ + éó′ + èó = Ñ (ñ)K==

=
qÜÉ=ÖÉåÉê~ä=ëçäìíáçå=áë=ÖáîÉå=Äó==
ó = óé + óÜ I=ïÜÉêÉ==
óé =áë==~=é~êíáÅìä~ê==ëçäìíáçå=çÑ==íÜÉ=áåÜçãçÖÉåÉçìë=Éèì~íáçå=
~åÇ= óÜ =áë=íÜÉ=ÖÉåÉê~ä=ëçäìíáçå=çÑ=íÜÉ=~ëëçÅá~íÉÇ=ÜçãçÖÉåÉ-
çìë=Éèì~íáçå=EëÉÉ=íÜÉ=éêÉîáçìë=íçéáÅ=NNTPFK=
=
fÑ=íÜÉ=êáÖÜí=ëáÇÉ=Ü~ë=íÜÉ=Ñçêã==

Ñ (ñ) = Éαñ (mN(ñ)Åçëβñ + mN(ñ)ëáåβñ)I==

íÜÉå=íÜÉ==é~êíáÅìä~ê=ëçäìíáçå= óé =áë=ÖáîÉå=Äó==

óé = ñ âÉαñ (oN(ñ)Åçëβñ + oO(ñ)ëáåβñ) I==
ïÜÉêÉ= íÜÉ= éçäóåçãá~äë= oN(ñ)= ~åÇ= oO(ñ)= Ü~îÉ= íç= ÄÉ= ÑçìåÇ=

Äó=ìëáåÖ=íÜÉ=ãÉíÜçÇ=çÑ=ìåÇÉíÉêãáåÉÇ=ÅçÉÑÑáÅáÉåíëK==
• fÑ= α + βá =áë=åçí=~=êççí=çÑ=íÜÉ=ÅÜ~ê~ÅíÉêáëíáÅ=Éèì~íáçåI=íÜÉå=

íÜÉ=éçïÉê= â = M I=
• fÑ= α + βá =áë=~=ëáãéäÉ=êççíI=íÜÉå= â = NI=
• fÑ= α + βá =áë=~=ÇçìÄäÉ=êççíI=íÜÉå= â = O K==
=
1175. aáÑÑÉêÉåíá~ä=bèì~íáçåë=ïáíÜ=ó=jáëëáåÖ=

ó′′ = Ñ (ñIó′)K==

pÉí= ì = ó′ K=qÜÉå=íÜÉ=åÉï=Éèì~íáçå=ë~íáëÑáÉÇ=Äó=î=áë==

ì′ = Ñ (ñIì)I==

ïÜáÅÜ=áë=~=Ñáêëí=çêÇÉê=ÇáÑÑÉêÉåíá~ä=Éèì~íáçåK=

299

CHAPTER 10. DIFFERENTIAL EQUATIONS

1176. aáÑÑÉêÉåíá~ä=bèì~íáçåë=ïáíÜ=ñ=jáëëáåÖ=

ó′′ = Ñ (óIó′)K==

pÉí= ì = ó′ K=páåÅÉ==

ó′′ = Çì = Çì Çó = ì Çì I==
Çñ Çó Çñ Çó

ïÉ=Ü~îÉ=

ì Çì = Ñ (óIì) I==

Çó

ïÜáÅÜ=áë=~=Ñáêëí=çêÇÉê=ÇáÑÑÉêÉåíá~ä=Éèì~íáçåK==

=
1177. cêÉÉ=råÇ~ãéÉÇ=sáÄê~íáçåë=

qÜÉ=ãçíáçå=çÑ=~=j~ëë=çå=~=péêáåÖ=áë=ÇÉëÅêáÄÉÇ=Äó=íÜÉ=Éèì~-

íáçå==
ã&ó& + âó = M I==

ïÜÉêÉ==
ã=áë=íÜÉ=ã~ëë=çÑ=íÜÉ=çÄàÉÅíI=
â=áë=íÜÉ=ëíáÑÑåÉëë=çÑ=íÜÉ=ëéêáåÖI=
ó=áë=Çáëéä~ÅÉãÉåí=çÑ=íÜÉ=ã~ëë=Ñêçã=ÉèìáäáÄêáìãK=
=
qÜÉ=ÖÉåÉê~ä=ëçäìíáçå=áë==

ó = ^Åçë(ωMí − δ)I==

ïÜÉêÉ==
^=áë=íÜÉ=~ãéäáíìÇÉ=çÑ=íÜÉ=Çáëéä~ÅÉãÉåíI=

ωM =áë=íÜÉ=ÑìåÇ~ãÉåí~ä=ÑêÉèìÉåÅóI=íÜÉ=éÉêáçÇ=áë= q = Oπ I=
ωM

δ =áë=éÜ~ëÉ=~åÖäÉ=çÑ=íÜÉ=Çáëéä~ÅÉãÉåíK=

qÜáë=áë=~å=Éñ~ãéäÉ=çÑ=ëáãéäÉ=Ü~êãçåáÅ=ãçíáçåK==

=

1178. cêÉÉ=a~ãéÉÇ=sáÄê~íáçåë=

ã&ó& + γó& + âó = M I=ïÜÉêÉ==

γ =áë=íÜÉ=Ç~ãéáåÖ=ÅçÉÑÑáÅáÉåíK==

qÜÉêÉ=~êÉ=P=Å~ëÉë=Ñçê=íÜÉ=ÖÉåÉê~ä=ëçäìíáçåW=
=

300

CHAPTER 10. DIFFERENTIAL EQUATIONS

`~ëÉ=NK γO > Qâã =EçîÉêÇ~ãéÉÇF=

ó(í) = ^ÉλNí + _ÉλOí I==

ïÜÉêÉ==

λN = − γ − γO − Qâã I= λO = − γ + γO − Qâã K==
Oã Oã

=

`~ëÉ=OK= γO = Qâã EÅêáíáÅ~ääó=Ç~ãéÉÇF=

ó(í) = (^ + _í)Éλí I==

ïÜÉêÉ==

λ = − γ K=
Oã

=

`~ëÉ=PK= γO < Qâã =EìåÇÉêÇ~ãéÉÇF==

ó (í) = É − γ í ^ Åçë(ωí − δ) I=ïÜÉêÉ==
Oã

ω = Qâã − γO K==

=
1179. páãéäÉ=mÉåÇìäìã=

ÇOθ + Ö θ = M I=
ÇíO i

ïÜÉêÉ= θ = áë= íÜÉ= ~åÖìä~ê= Çáëéä~ÅÉãÉåíI= i= áë= íÜÉ= éÉåÇìäìã=

äÉåÖíÜI=Ö=áë=íÜÉ=~ÅÅÉäÉê~íáçå=çÑ=Öê~îáíóK=

=

qÜÉ=ÖÉåÉê~ä=ëçäìíáçå=Ñçê=ëã~ää=~åÖäÉë= θ =áë==

θ(í) = θã~ñ ëáå Ö í I=íÜÉ=éÉêáçÇ=áë= q = Oπ i K==
i Ö

=

1180. oi`=`áêÅìáí=

i ÇOf + o Çf + N f = s′(í) = ωbM Åçë(ωí) I=
ÇíO Çí `

301

CHAPTER 10. DIFFERENTIAL EQUATIONS

ïÜÉêÉ= f= áë= íÜÉ= ÅìêêÉåí= áå= ~å= oi`= ÅáêÅìáí= ïáíÜ= ~å= ~Å= îçäí~ÖÉ=

ëçìêÅÉ= s(í) = bM ëáå(ωí)K===

=
qÜÉ=ÖÉåÉê~ä=ëçäìíáçå=áë==

f(í) = `NÉêNí + `OÉêOí + ^ ëáå(ωí − ϕ)I=

ïÜÉêÉ==

− o ± oO − Qi
` I==
êNI O = Oi

^ = ωbM I==
O
 iωO − N  + oOωO
 `

ϕ = ~êÅí~å iω − N  I==
 o o`ω 

`NI= `O =~êÉ=Åçåëí~åíë=ÇÉéÉåÇáåÖ=çå=áåáíá~ä=ÅçåÇáíáçåëK=

=

=

=

10.3. Some Partial Differential Equations

=

1181. qÜÉ=i~éä~ÅÉ=bèì~íáçå=

∂Oì + ∂Oì = M =
∂ñ O ∂ó O

~ééäáÉë=íç=éçíÉåíá~ä=ÉåÉêÖó=ÑìåÅíáçå= ì(ñI ó)==Ñçê==~==ÅçåëÉê-

î~íáîÉ= ÑçêÅÉ= ÑáÉäÇ= áå= íÜÉ= ñó-éä~åÉK= m~êíá~ä= ÇáÑÑÉêÉåíá~ä= Éèì~-

íáçåë=çÑ=íÜáë=íóéÉ=~êÉ=Å~ääÉÇ=ÉääáéíáÅK==

=

1182. qÜÉ=eÉ~í=bèì~íáçå=

∂Oì + ∂Oì = ∂ì =
∂ñ O ∂ó O ∂í

302

CHAPTER 10. DIFFERENTIAL EQUATIONS

~ééäáÉë= íç= íÜÉ= íÉãéÉê~íìêÉ= ÇáëíêáÄìíáçå= ì(ñI ó)= áå= íÜÉ= ñó-

éä~åÉ=ïÜÉå=ÜÉ~í=áë=~ääçïÉÇ=íç=Ñäçï=Ñêçã=ï~êã=~êÉ~ë=íç=Åççä=

çåÉëK=qÜÉ=Éèì~íáçåë=çÑ=íÜáë=íóéÉ=~êÉ=Å~ääÉÇ=é~ê~ÄçäáÅK==

=
1183. qÜÉ=t~îÉ=bèì~íáçå=

∂Oì + ∂Oì = ∂Oì =
∂ñ O ∂ó O ∂íO

~ééäáÉë=íç=íÜÉ=Çáëéä~ÅÉãÉåí= ì(ñIó)=çÑ=îáÄê~íáåÖ=ãÉãÄê~åÉë=

~åÇ= çíÜÉê= ï~îÉ= ÑìåÅíáçåëK= qÜÉ= Éèì~íáçåë= çÑ= íÜáë= íóéÉ= ~êÉ=

Å~ääÉÇ=ÜóéÉêÄçäáÅK==

=

=

303

Chapter 11

Series

=
=
=
=

11.1 Arithmetic Series

=

fåáíá~ä=íÉêãW= ~N =

kíÜ=íÉêãW= ~å =
aáÑÑÉêÉåÅÉ=ÄÉíïÉÉå=ëìÅÅÉëëáîÉ=íÉêãëW=Ç=

kìãÄÉê=çÑ=íÉêãë=áå=íÜÉ=ëÉêáÉëW=å=

pìã=çÑ=íÜÉ=Ñáêëí=å=íÉêãëW= på =
=

=
1184. ~å = ~å−N + Ç = ~å−O + OÇ = K = ~N + (å −N)Ç =

=

1185. ~N + ~å = ~O + ~å−N = K = ~á + ~å+N−á =
=

1186. ~á = ~ á −N + ~á+N =
O

= (å − N)Ç

1187. på = ~N + ~å ⋅å = O~N + O ⋅å =
O

=

=

=

=

=

304

CHAPTER 11. SERIES

11.2 Geometric Series

=
fåáíá~ä=íÉêãW= ~N =

kíÜ=íÉêãW= ~å =

`çããçå=ê~íáçW=è=
kìãÄÉê=çÑ=íÉêãë=áå=íÜÉ=ëÉêáÉëW=å=
pìã=çÑ=íÜÉ=Ñáêëí=å=íÉêãëW= på =

pìã=íç=áåÑáåáíóW=p=
=
=

1188. ~å = è~å−N = ~Nèå−N =

=
1189. ~N ⋅ ~å = ~O ⋅ ~å−N = K = ~á ⋅ ~å+N−á =

=

1190. ~á = ~á−N ⋅ ~á+N =

=
( )1191.
på = ~åè − ~N = ~N èå −N =
è −N è −N

=

1192. p = äáã på = ~N =
N− è
å→∞

cçê= è < NI=íÜÉ=ëìã=p=ÅçåîÉêÖÉë=~ë= å → ∞ K=

=

=

=

11.3 Some Finite Series

=
kìãÄÉê=çÑ=íÉêãë=áå=íÜÉ=ëÉêáÉëW=å=
=
=

305

CHAPTER 11. SERIES

1193. N+ O + P + K + å = å(å + N) =

O

=

1194. O + Q + S +K+ Oå = å(å + N)=

=

1195. N+ P + R +K+ (Oå −N) = åO =

= å(Oâ + å −N)

1196. â + (â +N)+ (â + O)+K+ (â + å −N) = O =

= å(å N)(Oå N)

1197. NO + OO + PO +K + åO = + S + =

=

1198. NP + OP + PP +K+ åP = å(å + N)O =

 O 

=
( )1199.
NO + PO + RO +K+ (Oå −N)O = å QåO −N =
P

=

( )1200. NP + PP + RP +K+ (Oå −N)P = åO OåO −N =

=

1201. N+ N + N + N +K+ N +K= O=
O Q U Oå

=

1202. N + N + N +K+ N +K=N=
N⋅ O O⋅P P⋅Q
å(å +N)

=

1203. N+ N + N + N + K+ (å N N)> + K = É =
N> O> P> −

=

=

=

306

CHAPTER 11. SERIES

11.4 Infinite Series

=

pÉèìÉåÅÉW= {~ å }=

cáêëí=íÉêãW= ~N =
kíÜ=íÉêãW= ~å =
=

=

1204. fåÑáåáíÉ=pÉêáÉë=

∞

∑~å = ~N + ~O +K+ ~å +K=

å=N

=

1205. kíÜ=m~êíá~ä=pìã=

å

∑på = ~å = ~N + ~O +K + ~å =
å=N

=

1206. `çåîÉêÖÉåÅÉ=çÑ=fåÑáåáíÉ=pÉêáÉë=

∞
∑~å
= i I=áÑ= äáã på = i =
å=N
å→∞

=

1207. kíÜ=qÉêã=qÉëí=

∞
∑•
fÑ=íÜÉ=ëÉêáÉë= ~å áë=ÅçåîÉêÖÉåíI=íÜÉå= äáã ~å = M K==

å=N å→∞

• fÑ= äáã ~ å ≠ M I=íÜÉå=íÜÉ=ëÉêáÉë=áë=ÇáîÉêÖÉåíK=

å→∞

=

=

=

11.5 Properties of Convergent Series

=

∞∞

`çåîÉêÖÉåí=pÉêáÉëW= ∑~å = ^ I= ∑Äå = _ =
å=N å=N

oÉ~ä=åìãÄÉêW=Å=

307

CHAPTER 11. SERIES

∞ ∞∞

1208. ∑ (~å + Äå ) = ∑ ~å + ∑ Äå = ^ + _ =
å=N å=N å=N

=

∞∞

1209. ∑ Å~å = Å∑ ~å = Å^ K==
å=N å=N

=
=
=

11.6 Convergence Tests

=

1210. qÜÉ=`çãé~êáëçå=qÉëí=

∞∞

∑ ∑iÉí= ~å =~åÇ= Äå =ÄÉ=ëÉêáÉë=ëìÅÜ=íÜ~í= M < ~å ≤ Äå =Ñçê=~ää=åK==
å=N å=N

∞∞

∑ ∑• fÑ= Äå áë=ÅçåîÉêÖÉåí=íÜÉå= ~å áë=~äëç=ÅçåîÉêÖÉåíK==
å=N å=N

∞∞

∑ ∑• fÑ= ~å áë=ÇáîÉêÖÉåí=íÜÉå= Äå áë=~äëç=ÇáîÉêÖÉåíK=
å=N å=N

=

1211. qÜÉ=iáãáí=`çãé~êáëçå=qÉëí=

∞∞

∑ ∑iÉí= ~å =~åÇ= Äå =ÄÉ=ëÉêáÉë=ëìÅÜ=íÜ~í= ~å =~åÇ= Äå =~êÉ=éçëá-
å=N å=N

íáîÉ=Ñçê=~ää=åK==

∑ ∑• M< äáã ~å ∞ ∞
fÑ= Äå < ∞ = íÜÉå= = ~åÇ= ~êÉ= ÉáíÜÉê= ÄçíÜ=
å→∞ ~å Äå

å=N å=N

ÅçåîÉêÖÉåí=çê=ÄçíÜ=ÇáîÉêÖÉåíK=

∑ ∑• ~å ∞ ∞
fÑ= äáã Äå = M =íÜÉå= ÅçåîÉêÖÉåí=áãéäáÉë=íÜ~í= =áë=
å→∞ Äå ~å

å=N å=N

~äëç=ÅçåîÉêÖÉåíK=

308

CHAPTER 11. SERIES

∑ ∑• äáã ~å ∞ ∞
fÑ= Äå = ∞ =íÜÉå= ÇáîÉêÖÉåí=áãéäáÉë=íÜ~í= =áë=
å→∞ Äå ~å

å=N å=N

~äëç=ÇáîÉêÖÉåíK=

=
1212. é-ëÉêáÉë=

∑é-ëÉêáÉë= ∞ N =ÅçåîÉêÖÉë=Ñçê= é > N=~åÇ=ÇáîÉêÖÉë=Ñçê=
å=N åé

M < é ≤ NK=

=
1213. qÜÉ=fåíÉÖê~ä=qÉëí=

iÉí= Ñ (ñ)= ÄÉ= ~= ÑìåÅíáçå= ïÜáÅÜ= áë= ÅçåíáåìçìëI= éçëáíáîÉI= ~åÇ=

ÇÉÅêÉ~ëáåÖ=Ñçê=~ää= ñ ≥ NK=qÜÉ=ëÉêáÉë==

∞

∑Ñ (å) = Ñ (N)+ Ñ (O)+ Ñ (P)+K+ Ñ (å)+K=

å=N

∞

ÅçåîÉêÖÉë=áÑ= ∫ Ñ (ñ)Çñ ÅçåîÉêÖÉëI=~åÇ=ÇáîÉêÖÉë=áÑ=
N

å

∫ Ñ (ñ)Çñ → ∞ =~ë= å → ∞ K=

N

=
1214. qÜÉ=o~íáç=qÉëí=

∞

iÉí= ∑~å =ÄÉ=~=ëÉêáÉë=ïáíÜ=éçëáíáîÉ=íÉêãëK=
å=N

∑• fÑ= ~ å +N ∞
äáã ~å < N =íÜÉå= å=N ~å áë=ÅçåîÉêÖÉåíK=

å→∞

∑• ~ å +N ∞
fÑ= äáã ~å > N=íÜÉå= å=N ~å =áë=ÇáîÉêÖÉåíK=

å→∞

∑• fÑ= äáã ~ å +N ∞
~å = N= íÜÉå= = ã~ó= ÅçåîÉêÖÉ= çê= ÇáîÉêÖÉ= ~åÇ=
å→∞ ~å

å=N

íÜÉ= ê~íáç= íÉëí= áë= áåÅçåÅäìëáîÉX= ëçãÉ= çíÜÉê= íÉëíë= ãìëí= ÄÉ=

ìëÉÇK==

=

309

CHAPTER 11. SERIES

1215. qÜÉ=oççí=qÉëí=

∞

iÉí= ∑~å =ÄÉ=~=ëÉêáÉë=ïáíÜ=éçëáíáîÉ=íÉêãëK=
å=N

∞
∑•
fÑ= äáã å ~å < N=íÜÉå= ~å =áë=ÅçåîÉêÖÉåíK=
å→∞
å=N

∞
∑•
fÑ= äáã å ~å > N=íÜÉå= ~å =áë=ÇáîÉêÖÉåíK=
å→∞
å=N

∞
∑•
fÑ= äáã å ~å = N=íÜÉå= ~å = ã~ó= ÅçåîÉêÖÉ= çê= ÇáîÉêÖÉI= Äìí=

å→∞ å=N

åç=ÅçåÅäìëáçå=Å~å=ÄÉ=Çê~ïå=Ñêçã=íÜáë=íÉëíK=

=

=

=

11.7 Alternating Series

=

1216. qÜÉ=^äíÉêå~íáåÖ=pÉêáÉë=qÉëí=EiÉáÄåáò∞ë=qÜÉçêÉãF=

= iÉí= {~ å }=ÄÉ=~=ëÉèìÉåÅÉ=çÑ=éçëáíáîÉ=åìãÄÉêë=ëìÅÜ=íÜ~í=

~å+N < ~å =Ñçê=~ää=åK=

äáã ~ å = M K==

å→∞

∞∞
(−N)å ~å = ~åÇ= (− )N~å−N
∑ ∑qÜÉå= íÜÉ= ~äíÉêå~íáåÖ= ëÉêáÉë= å =

å=N å=N

ÄçíÜ=ÅçåîÉêÖÉK====

=

1217. ^ÄëçäìíÉ=`çåîÉêÖÉåÅÉ=

∞

∑• ^= ëÉêáÉë= ~å = áë= ~ÄëçäìíÉäó= ÅçåîÉêÖÉåí= áÑ= íÜÉ= ëÉêáÉë=
å=N

∞

∑ ~å =áë=ÅçåîÉêÖÉåíK==

å=N

310

CHAPTER 11. SERIES

∞

∑• fÑ=íÜÉ=ëÉêáÉë= ~å áë=~ÄëçäìíÉäó=ÅçåîÉêÖÉåí=íÜÉå=áí=áë=Åçå-
å=N
îÉêÖÉåíK=

=
1218. `çåÇáíáçå~ä=`çåîÉêÖÉåÅÉ=

∞

^= ëÉêáÉë= ∑~å áë= ÅçåÇáíáçå~ääó= ÅçåîÉêÖÉåí= áÑ= íÜÉ= ëÉêáÉë= áë=
å=N

ÅçåîÉêÖÉåí=Äìí=áë=åçí=~ÄëçäìíÉäó=ÅçåîÉêÖÉåíK=
=
=
=

11.8 Power Series

=
oÉ~ä=åìãÄÉêëW=ñI= ñM =

∞∞

∑ ∑mçïÉê=ëÉêáÉëW= ~åñå I= ~å (ñ − ñM )å =
å=M å=M

tÜçäÉ=åìãÄÉêW=å=
o~Çáìë=çÑ=`çåîÉêÖÉåÅÉW=o=
=
1219. mçïÉê=pÉêáÉë=áå=ñ=

∞

∑~åñå = ~M + ~Nñ + ~OñO +K+ ~åñå +K=

å=M

=

1220. mçïÉê=pÉêáÉë=áå= (ñ − ñM ) =

∞

∑ ~å (ñ − ñ M )å = ~M + ~N (ñ − ñ M ) + ~O (ñ − ñ M )O + K + ~å (ñ − ñ M )å + K

å=M

=
1221. fåíÉêî~ä=çÑ=`çåîÉêÖÉåÅÉ===

qÜÉ=ëÉí=çÑ=íÜçëÉ=î~äìÉë=çÑ=ñ=Ñçê=ïÜáÅÜ=íÜÉ=ÑìåÅíáçå=

∞

Ñ (ñ) = ∑~å (ñ − ñM )å =áë=ÅçåîÉêÖÉåí=áë=Å~ääÉÇ==íÜÉ==áåíÉêî~ä=çÑ=
å=M

ÅçåîÉêÖÉåÅÉK=

311

CHAPTER 11. SERIES

1222. o~Çáìë=çÑ=`çåîÉêÖÉåÅÉ=

fÑ=íÜÉ=áåíÉêî~ä=çÑ=ÅçåîÉêÖÉåÅÉ=áë== (ñM − oI ñM + o) ==Ñçê==ëçãÉ=

o ≥ M I=íÜÉ=o=áë=Å~ääÉÇ==íÜÉ=ê~Çáìë=çÑ=ÅçåîÉêÖÉåÅÉK==fí=áë=ÖáîÉå=
~ë=

o = äáã N =çê= o = äáã ~å K==
~å→∞ å ~å→∞
å å+N

=

=

=

11.9 Differentiation and Integration of Power

Series

=

`çåíáåìçìë=ÑìåÅíáçåW= Ñ (ñ)=

∞

∑mçïÉê=ëÉêáÉëW= ~åñå =
å=M

tÜçäÉ=åìãÄÉêW=å=
o~Çáìë=çÑ=`çåîÉêÖÉåÅÉW=o=
=
=
1223. aáÑÑÉêÉåíá~íáçå=çÑ=mçïÉê=pÉêáÉë=

∞

∑iÉí= Ñ (ñ) = ~åñå = ~M + ~Nñ + ~OñO +K=Ñçê= ñ < o K==
å=M

qÜÉåI= = Ñçê= ñ < o I= Ñ (ñ)= áë= ÅçåíáåìçìëI= íÜÉ= ÇÉêáî~íáîÉ= Ñ′(ñ)=

Éñáëíë=~åÇ=

Ñ ′(ñ ) = Ç ~M + Ç ~Nñ + Ç ~Oñ O + K=
Çñ Çñ Çñ

∞

∑= ~N + O~Oñ + P~PñO + K = å~åñå−N K=
å=N

=

=

=

312

CHAPTER 11. SERIES

1224. fåíÉÖê~íáçå=çÑ=mçïÉê=pÉêáÉë=

∞

∑iÉí= Ñ (ñ) = ~åñå = ~M + ~Nñ + ~OñO +K=Ñçê= ñ < o K==
å=M

qÜÉåI==Ñçê= ñ < o I=íÜÉ=áåÇÉÑáåáíÉ=áåíÉÖê~ä= ∫ Ñ (ñ)Çñ Éñáëíë=~åÇ==

∫ Ñ (ñ)Çñ = ∫ ~MÇñ + ∫ ~NñÇñ + ∫ ~OñOÇñ +K =

∑= ~Mñ ñO ñP ∞ ñ å+N + ` K=
+ ~N O + ~O P +K= å+N
~å

å=M

=

=

=

11.10 Taylor and Maclaurin Series

=

tÜçäÉ=åìãÄÉêW=å=

aáÑÑÉêÉåíá~ÄäÉ=ÑìåÅíáçåW= Ñ (ñ)=

oÉã~áåÇÉê=íÉêãW= oå =
=

=
1225. q~óäçê=pÉêáÉë=

Ñ(ñ) = ∞ Ñ (å ) (~ ) (ñ − ~)å = Ñ (~)+ Ñ ′(~)(ñ − ~)+ Ñ ′′(~)(ñ − ~)O +K

∑ å> O>
å=M

== + Ñ (å)(~)(ñ − ~)å + oå K==

å>

=
1226. qÜÉ=oÉã~áåÇÉê=^ÑíÉê=åHN=qÉêãë=áë=ÖáîÉå=Äó==

oå = Ñ (å+N)(ξ)(ñ − )~ å+N I=== ~ < ξ < ñ K=
(å +N)>

=
1227. j~Åä~ìêáå=pÉêáÉë=

313

CHAPTER 11. SERIES

∑Ñ(ñ) = ∞ Ñ (å) (M) ñ å = Ñ (M)+ Ñ ′(M)ñ + Ñ ′′(M)ñO +K+ Ñ (å)(M)ñ å + oå

å=M å> O> å>

=

=

=

=

11.11 Power Series Expansions for Some

Functions

=

tÜçäÉ=åìãÄÉêW=å=

oÉ~ä=åìãÄÉêW=ñ=

=

=

1228. Éñ =N+ ñ + ñO + ñP +K+ ñå +K=
O> P> å>

=

1229. ~ñ =N+ ñ äå ~ + (ñ äå ~)O + (ñ äå~)P +K+ (ñ äå ~)å +K=
N>
O> P> å>

=

1230. äå(N+ ñ)= ñ − ñO + ñP − ñQ +K+ (− N)å ñå+N ±KI= −N< ñ ≤ NK=
O P Q
å+N

=

1231. äå N+ ñ = O ñ + ñP + ñR + ñT + K I= ñ < N K=
N− ñ P R T

=

1232. äå ñ =  ñ − N + N  ñ − N P + N  ñ − N R  I= ñ > M K=
O ñ + N P  ñ + N  R  ñ + N  K



=

1233. Åçë ñ =N− ñO + ñQ − ñS +K+ (− N)å ñOå ±K=
O> Q> S> (Oå)>

=

314

CHAPTER 11. SERIES

1234. ëáå ñ = ñ − ñP + ñR − ñT +K+ ( )− N å ñOå+N ±K=
P> R> T> (Oå +N)>

=

1235. í~å ñ = ñ + ñP + Oñ R + NT ñ T + SOñ V + K I= ñ < π K=
P NR PNR OUPR O

=

1236. Åçí ñ = N −  ñ + ñP + Oñ R + Oñ T +K I= ñ < π K=
ñ P QR VQR QTOR

=

1237. ~êÅëáå ñ = ñ + ñP + N⋅Pñ R +K+ N⋅ P ⋅ RK(Oå − N)ñ Oå+N + K I=
O⋅P O⋅Q⋅R O ⋅ Q ⋅ SK(Oå)(Oå + N)

ñ < N K=

=

1238. ~êÅÅçë ñ = π −  ñ + ñP + N⋅ Pñ R +K+ N⋅ P ⋅ RK(Oå − N)ñ Oå+N + KI
O O⋅P O⋅Q⋅R O ⋅ Q ⋅ SK(Oå)(Oå + N)

ñ < N K=

=

1239. ~êÅí~å ñ = ñ − ñP + ñR − ñT +K+ ( )− N å ñ Oå+N ±K I= ñ ≤ N K=
P R T
Oå + N

=

1240. ÅçëÜ ñ =N+ ñO + ñQ + ñS +K+ ñ Oå +K=
O> Q> S>
(Oå)>

=

1241. ëáåÜ ñ =ñ + ñP + ñR + ñT +K+ ñ Oå+N +K=
P> R> T>
(Oå + N)>

=
=
=
=
=

315

CHAPTER 11. SERIES

11.12 Binomial Series

=

tÜçäÉ=åìãÄÉêëW=åI=ã=

oÉ~ä=åìãÄÉêW=ñ=

`çãÄáå~íáçåëW= å `ã =
=

=

1242. (N+ ñ)å = N+ å`Nñ + å`OñO + K + ã`åñã + K + ñå =

= å(å −N)K[å − (ã −N)] I=

1243. å `ã = ã> ñ < N K=

=

1244. N =N− ñ + ñO − ñP + K I= ñ < N K=
N+ ñ

=

1245. N =N+ ñ + ñO + ñP + K I= ñ < N K=
N− ñ

=

1246. N+ ñ =N+ ñ − ñO + N⋅ Pñ P − N⋅ P ⋅ Rñ Q + K I= ñ ≤ N K=
O O⋅Q O⋅Q⋅S O⋅Q⋅S⋅U

=

1247. P N+ ñ =N+ ñ − N⋅ OñO + N⋅ O⋅RñP − N⋅ O⋅R⋅UñQ +K I= ñ ≤ N K=
P P⋅S P⋅S⋅V P⋅ S ⋅ V ⋅NO

=

=

=

11.13 Fourier Series

=

fåíÉÖê~ÄäÉ=ÑìåÅíáçåW= Ñ (ñ)=

cçìêáÉê=ÅçÉÑÑáÅáÉåíëW= ~M I= ~å I= Äå =
tÜçäÉ=åìãÄÉêW=å==

316

CHAPTER 11. SERIES

Ñ(ñ) = ~M ∞ ëáå åñ ) =
∑1248. O + Åçë åñ + Äå
(~å

å=N

=

∫1249.~å= N π Çñ ==
π
Ñ (ñ)Åçëåñ

−π

=

∫1250.Äå= N π Çñ ==
π
Ñ (ñ)ëáååñ

−π

=

=

317

Chapter 12

Probability

=
=
=
=

12.1 Permutations and Combinations

=

mÉêãìí~íáçåëW= å mã =

`çãÄáå~íáçåëW= å `ã =
tÜçäÉ=åìãÄÉêëW=åI=ã=

=

=

1251. c~Åíçêá~ä=

å>= N⋅O⋅PK(å − O)(å −N)å =

M>= N=

=

1252. å må = å> =
=

1253. å mã = (å å> =

− ã)>

=

1254. _áåçãá~ä=`çÉÑÑáÅáÉåí=

å `ã = ãå  = å> ã)> =

ã>(å −

=

1255. å `ã =å `å−ã =
=

1256. å `ã +å `ã+N =å+N `ã+N =
=

318

CHAPTER 12. PROBABILITY

1257. å `M +å `N +å `O + K +å `å = Oå =
=

1258. m~ëÅ~ä∞ë=qêá~åÖäÉ=
=

oçï=M= = = = = = = N= = = = = = =
oçï=N= = = = = = N= = N= = = = = =
oçï=O= = = = = N= = O= = N= = = = =
oçï=P= = = = N= = P= = P= = N= = = =
oçï=Q= = = N= = Q= = S= = Q= = N= = =
oçï=R= = N= = R= = NM= = NM= = R= = N= =
oçï=S= N= = S= = NR= = OM= = NR= = S= = N=

=
=
=

12.2 Probability Formulas

=

bîÉåíëW=^I=_=

mêçÄ~ÄáäáíóW=m=

o~åÇçã=î~êá~ÄäÉëW=uI=vI=w=

s~äìÉë=çÑ=ê~åÇçã=î~êá~ÄäÉëW=ñI=óI=ò=

bñéÉÅíÉÇ=î~äìÉ=çÑ=uW= µ =

^åó=éçëáíáîÉ=êÉ~ä=åìãÄÉêW= ε ==
pí~åÇ~êÇ=ÇÉîá~íáçåW= σ =

s~êá~åÅÉW= σO =

aÉåëáíó=ÑìåÅíáçåëW= Ñ (ñ)I= Ñ (í)=

=

=
1259. mêçÄ~Äáäáíó=çÑ=~å=bîÉåí=

m(^) = ã I==
å

ïÜÉêÉ==

ã=áë=íÜÉ=åìãÄÉê=çÑ=éçëëáÄäÉ=éçëáíáîÉ=çìíÅçãÉëI==

å=áë=íÜÉ=íçí~ä=åìãÄÉê=çÑ=éçëëáÄäÉ=çìíÅçãÉëK=

=

319

CHAPTER 12. PROBABILITY

1260. o~åÖÉ=çÑ=mêçÄ~Äáäáíó=s~äìÉë=

M ≤ m(^) ≤ N=

=
1261. `Éêí~áå=bîÉåí=

m(^) = N=

=
1262. fãéçëëáÄäÉ=bîÉåí=

m(^) = M =

=
1263. `çãéäÉãÉåí=

m(^)= N− m(^) =

=
1264. fåÇÉéÉåÇÉåí=bîÉåíë=

m(^L_) = m(^)I==

m(_L ^) = m(_) =

=
1265. ^ÇÇáíáçå=oìäÉ=Ñçê=fåÇÉéÉåÇÉåí=bîÉåíë=

m(^ ∪ _) = m(^) + m(_)=

=
1266. jìäíáéäáÅ~íáçå=oìäÉ=Ñçê=fåÇÉéÉåÇÉåí=bîÉåíë=

m(^ ∩ _) = m(^)⋅ m(_)=

=
1267. dÉåÉê~ä=^ÇÇáíáçå=oìäÉ=

m(^ ∪ _) = m(^)+ m(_)− m(^ ∩ _)I==

ïÜÉêÉ==

^ ∪ _ =áë=íÜÉ=ìåáçå=çÑ=ÉîÉåíë=^=~åÇ=_I==

^ ∩ _ =áë=íÜÉ=áåíÉêëÉÅíáçå=çÑ=ÉîÉåíë=^=~åÇ=_K=

=

1268. `çåÇáíáçå~ä=mêçÄ~Äáäáíó=
m(^ ∩ _)
m(^ L _) = m(_) =

=

1269. m(^ ∩ _) = m(_)⋅ m(^ L _) = m(^)⋅ m(_ L ^) =

320

CHAPTER 12. PROBABILITY

1270. i~ï=çÑ=qçí~ä=mêçÄ~Äáäáíó=

ã

m(^) = ∑m(_á )m(^ L_á ) I==
á=N

ïÜÉêÉ= _á =áë=~=ëÉèìÉåÅÉ=çÑ=ãìíì~ääó=ÉñÅäìëáîÉ=ÉîÉåíëK==

=

1271. _~óÉë∞=qÜÉçêÉã=
m(^ L _)⋅ m(_)
m(_ L ^) = m(^) =

=

1272. _~óÉë∞=cçêãìä~=
m(_á )⋅ m(^ L _á )
m(_á L ^) = I==
ã
∑ m(_á )⋅ m(^ L _á )
â =N

ïÜÉêÉ==

_á =áë=~=ëÉí=çÑ=ãìíì~ääó=ÉñÅäìëáîÉ=ÉîÉåíë=EÜóéçíÜÉëÉëFI=

^ =áë=íÜÉ=Ñáå~ä=ÉîÉåíI==
m(_á )=~êÉ=íÜÉ=éêáçê=éêçÄ~ÄáäáíáÉëI=
m(_á L ^) =~êÉ=íÜÉ=éçëíÉêáçê=éêçÄ~ÄáäáíáÉëK=

=

1273. i~ï=çÑ=i~êÖÉ=kìãÄÉêë=

m på − µ ≥ ε → M =~ë= å → ∞ I==
å 

m på − µ < ε → N =~ë= å → ∞ I==
å 

ïÜÉêÉ==

på =áë=íÜÉ=ëìã=çÑ=ê~åÇçã=î~êá~ÄäÉëI=

å =áë=íÜÉ=åìãÄÉê=çÑ=éçëëáÄäÉ=çìíÅçãÉëK=

=

1274. `ÜÉÄóëÜÉî=fåÉèì~äáíó=
s(u)
m( u − µ ≥ ε)≤ I==
εO
ïÜÉêÉ= s(u) =áë=íÜÉ=î~êá~åÅÉ=çÑ=uK=

321

CHAPTER 12. PROBABILITY

1275. kçêã~ä=aÉåëáíó=cìåÅíáçå=

N ( ñ −µ ) O

( )ϕ ñ = É− OσO I==

σ Oπ

ïÜÉêÉ=ñ=áë=~=é~êíáÅìä~ê=çìíÅçãÉK=

=
1276. pí~åÇ~êÇ=kçêã~ä=aÉåëáíó=cìåÅíáçå=

ϕ(ò) = N É−ò O =
O
Oπ

^îÉê~ÖÉ=î~äìÉ= µ = M I=ÇÉîá~íáçå= σ = NK=

=

===== =
=

Figure 210.

=

1277. pí~åÇ~êÇ=w=s~äìÉ=

w= u−µ =
σ

=

1278. `ìãìä~íáîÉ=kçêã~ä=aáëíêáÄìíáçå=cìåÅíáçå=

( ) ∫c ñ = N ñ (í−µ) O

É − OσO Çí I==

σ Oπ −∞

322

CHAPTER 12. PROBABILITY

ïÜÉêÉ==
ñ=áë=~=é~êíáÅìä~ê=çìíÅçãÉI==
í =áë=~=î~êá~ÄäÉ=çÑ=áåíÉÖê~íáçåK=
=

1279. m(α < u < β) = c α−µ  − c β −µ  I=
 σ   σ 

ïÜÉêÉ=

u=áë=åçêã~ääó=ÇáëíêáÄìíÉÇ=ê~åÇçã=î~êá~ÄäÉI=
c=áë=Åìãìä~íáîÉ=åçêã~ä=ÇáëíêáÄìíáçå=ÑìåÅíáçåI==

m(α < u < β) =áë=áåíÉêî~ä=éêçÄ~ÄáäáíóK=

=

1280. m( u − µ < ε)= Oc ε  I==
 σ 

ïÜÉêÉ==

u=áë=åçêã~ääó=ÇáëíêáÄìíÉÇ=ê~åÇçã=î~êá~ÄäÉI=
c=áë=Åìãìä~íáîÉ=åçêã~ä=ÇáëíêáÄìíáçå=ÑìåÅíáçåK=
=
1281. `ìãìä~íáîÉ=aáëíêáÄìíáçå=cìåÅíáçå=

ñ

c(ñ) = m(u < ñ) = ∫ Ñ (í)Çí I==
−∞

ïÜÉêÉ=í=áë=~=î~êá~ÄäÉ=çÑ=áåíÉÖê~íáçåK=

=
1282. _Éêåçìääá=qêá~äë=mêçÅÉëë=

µ = åé = I= σO = åéè I==

ïÜÉêÉ==
å =áë=~=ëÉèìÉåÅÉ=çÑ=ÉñéÉêáãÉåíëI==
é =áë=íÜÉ=éêçÄ~Äáäáíó=çÑ=ëìÅÅÉëë=çÑ=É~ÅÜ=ÉñéÉêáãÉåíëI=

è =áë=íÜÉ=éêçÄ~Äáäáíó=çÑ=Ñ~áäìêÉI= è = N− é K=

=
1283. _áåçãá~ä=aáëíêáÄìíáçå=cìåÅíáçå=======

Ä(åI éI è ) =  å  éâ è å−â I==
â

323

CHAPTER 12. PROBABILITY

µ = åé I= σO = åéè I=

Ñ (ñ) = (è + éÉñ )å I==

ïÜÉêÉ==
å=áë=íÜÉ=åìãÄÉê=çÑ=íêá~äë=çÑ=ëÉäÉÅíáçåëI=
é=áë=íÜÉ=éêçÄ~Äáäáíó=çÑ=ëìÅÅÉëëI=
è=áë=íÜÉ=éêçÄ~Äáäáíó=çÑ=Ñ~áäìêÉI= è = N− é K=

=
1284. dÉçãÉíêáÅ=aáëíêáÄìíáçå=

m(q = à) = èà−Né I==

µ = N I= σO = è I==
é éO

ïÜÉêÉ==
q=áë=íÜÉ=Ñáêëí=ëìÅÅÉëëÑìä=ÉîÉåí=áë=íÜÉ=ëÉêáÉëI=
à=áë=íÜÉ=ÉîÉåí=åìãÄÉêI=
é=áë=íÜÉ=éêçÄ~Äáäáíó=íÜ~í=~åó=çåÉ=ÉîÉåí=áë=ëìÅÅÉëëÑìäI==
è=áë=íÜÉ=éêçÄ~Äáäáíó=çÑ=Ñ~áäìêÉI= è = N− é K=

=
1285. mçáëëçå=aáëíêáÄìíáçå=

m(u = â) ≈ λâ É−λ I= λ = åé I==

â>

µ = λ I= σO = λ I==

ïÜÉêÉ==
λ =áë=íÜÉ=ê~íÉ=çÑ=çÅÅìêêÉåÅÉI=
â=áë=íÜÉ=åìãÄÉê=çÑ=éçëáíáîÉ=çìíÅçãÉëK=

=
1286. aÉåëáíó=cìåÅíáçå==

Ä

m(~ ≤ u ≤ Ä) = ∫ Ñ (ñ)Çñ ==
~

=
1287. `çåíáåìçìë=råáÑçêã=aÉåëáíó=

Ñ = N I= µ = ~ + Ä I==
− O
Ä ~

324

CHAPTER 12. PROBABILITY

ïÜÉêÉ=Ñ=áë=íÜÉ=ÇÉåëáíó=ÑìåÅíáçåK=
=
1288. bñéçåÉåíá~ä=aÉåëáíó=cìåÅíáçå=

Ñ (í) = λÉ−λí I= µ = λ I= σO = λO =

ïÜÉêÉ=í=áë=íáãÉI= λ =áë=íÜÉ=Ñ~áäìêÉ=ê~íÉK=
=
1289. bñéçåÉåíá~ä=aáëíêáÄìíáçå=cìåÅíáçå=

c(í) = N− É−λí I==

ïÜÉêÉ=í=áë=íáãÉI= λ =áë=íÜÉ=Ñ~áäìêÉ=ê~íÉK=
=
1290. bñéÉÅíÉÇ=s~äìÉ=çÑ=aáëÅêÉíÉ=o~åÇçã=s~êá~ÄäÉë=

å

µ = b(u) = ∑ ñáéá I==
á=N

ïÜÉêÉ= ñ á =áë=~=é~êíáÅìä~ê=çìíÅçãÉI= éá =áë=áíë=éêçÄ~ÄáäáíóK=
=
1291. bñéÉÅíÉÇ=s~äìÉ=çÑ=`çåíáåìçìë=o~åÇçã=s~êá~ÄäÉë=

∞

µ = b(u) = ∫ ñÑ(ñ)Çñ ==
−∞

=
1292. mêçéÉêíáÉë=çÑ=bñéÉÅí~íáçåë=

b(u + v) = b(u)+ b(v) I==
b(u − v) = b(u)− b(v)I=
b(Åu) = Åb(u) I==
b(uv) = b(u)⋅ b(v) I==

ïÜÉêÉ=Å=áë=~=Åçåëí~åíK=
=

( )1293. b uO = s(u)+ µO I==

ïÜÉêÉ==

µ = b(u)=áë=íÜÉ=ÉñéÉÅíÉÇ=î~äìÉI===
s(u) =áë=íÜÉ=î~êá~åÅÉK=

=
=
=

325

CHAPTER 12. PROBABILITY

1294. j~êâçî=fåÉèì~äáíó=

m(u > â)≤ b(u) I==

â
ïÜÉêÉ=â=áë=ëçãÉ=Åçåëí~åíK=
=
1295. s~êá~åÅÉ=çÑ=aáëÅêÉíÉ=o~åÇçã=s~êá~ÄäÉë=

[ ]å

σO = s(u) = b (u − µ)O ∑= (ñá − µ)O éá I==
á=N

ïÜÉêÉ==
ñ á =áë=~=é~êíáÅìä~ê=çìíÅçãÉI=
éá =áë=áíë=éêçÄ~ÄáäáíóK=
=
1296. s~êá~åÅÉ=çÑ=`çåíáåìçìë=o~åÇçã=s~êá~ÄäÉë=

[ ]∞

σO = s(u) = b (u − µ)O = ∫ (ñ − µ)OÑ (ñ)Çñ ==
−∞

=
1297. mêçéÉêíáÉë=çÑ=s~êá~åÅÉ=

s(u + v) = s(u)+ s(v) I==
s(u − v) = s(u)+ s(v) I=
s(u + Å) = s(u)I=
s(Åu) = ÅOs(u) I=

ïÜÉêÉ=Å=áë=~=Åçåëí~åíK=
=
1298. pí~åÇ~êÇ=aÉîá~íáçå=

[ ]a(u) = s(u) = b (u − µ)O =

=
1299. `çî~êá~åÅÉ=

Åçî(uIv) = b[(u − µ(u))(v − µ(v))]= b(uv)− µ(u)µ(v)I==

ïÜÉêÉ==
u =áë=ê~åÇçã=î~êá~ÄäÉI==

s(u) =áë=íÜÉ=î~êá~åÅÉ=çÑ=uI==

µ =áë=íÜÉ=ÉñéÉÅíÉÇ=î~äìÉ=çÑ=u=çê=vK=

326

CHAPTER 12. PROBABILITY

=

1300. `çêêÉä~íáçå=
Åçî(uI v )
ρ(uIv) = s(u)s(v ) I==

ïÜÉêÉ==

s(u) =áë=íÜÉ=î~êá~åÅÉ=çÑ=uI==

s(v) =áë=íÜÉ=î~êá~åÅÉ=çÑ=vK=

=
=
=

=
=

=

=
=

=

=
=

=
=
=

=
=

=

=
=

=

=
=

=
=
=

=
=
=

327

=
iççâ=Ñçê=çíÜÉê=Ü~åÇÄççâë=~åÇ=ëçäîÉÇ=éêçÄäÉã=ÖìáÇÉë=~í=
ïïïKã~íÜ-ÉÄççâëKÅçãK=
=

=
=
=