MACLAURIN THEOREM

MACLAURIN THEOREM & MACLAURIN SERIES
(PAST YEAR QUESTIONS)

1. Given that ln y​ ​ = sin–​ 1​ ​(2​x​), prove that (1 – 4​x2​ )​ – 4​x​– 4y​ ​ = 0.
Hence find Maclaurin’s series for y​ ,​ up to and including the term in x​ 3​ ​. [5]

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2. Given that y​ ​ = 1 + sin2​ ​ x​ ,​ show that + 4​y​ = 6. [3]
Hence, find the Maclaurin’s series for ​y,​ up to and including the term in x​ 3​ .​ [4]
Verify this result by using the expansion of sin ​x.​ [3]

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3. Given that y​ 2​ ​ = 1 + sin​2​ ​x​, show that y​ ​+ = 3 – 2​y​2​. [4]
Hence, find the Maclaurin’s series for ​y,​ up to and including the term in x​ 4​ .​ [6]
Verify this result by using the expansions of sin x​ ​ and . [5](76)

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4. If y​ ​ = (sin–​ 1 x​ )​ 2​ ,​ show that (1 – ​x2​ ​)– ​x​= 2. [3]
Hence, find the Maclaurin’s series for y​ ​, up to and including the term in ​x​6.​ [6]
By using a suitable value of x​ ​, use the expansion to estimate p​ 2​ ​ correct to
two decimal places. [3](90)

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5. Express lnas a power series in terms of x​ ​ up to the term in x​ 5​ ​.[3]
Hence, by using a suitable value of ​x​, estimate ln 3 correct to 3 d.p. [3](93)

6. By using the expansion of, show that for –1 < x​ ​ < 1,= – ln (1 – x​ ​). [4]
Hence, deduce the value of correct to four decimal places. [4](94)

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7. Given that d²y/dx²= x​ ​2​y​ + ​x,​ where y​ ​ = dy/dx = 1 when x​ ​ = 0. Obtain the
Maclaurin’s expansion for ​y​ up to and including the term in x​ 4​ ​. [5](96)

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8. By using the Maclaurin’s Theorem, show that the expansion of is
1 + x​ ​ + x​ 2​ ​ + .​ . .​ + ​x​r​ + ​. . . .

State the range of values of x​ ​ for this expansion valid. [6]
(a) By differentiating the series of , obtain the series for .

Hence, find the infinite series of
1 + 2( 1/2) + 3( 1/2 )²+ ​. . .​ + ​r (1/2) r − 1 + .​ . .​ [3]

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(b) By integrating the series of , obtain the series of ln (1 – ​x)​ .
Hence, find the infinite series of
1 + 1/2( 1/2) + 1/3( 1/2 )²+ ​. . .​ +1/ ​r (1/2) r − 1 + .​ . .​ [3][97]

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