Chapter 8: Q-18E (page 434)

Question 18: In Problems, find a power series expansion for $f (X)$ , given the expansion for f(x).
role="math" localid="1664283848012" $f (x) = \sin x = \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k}}{(2 k + 1)!} x^{2 k + 1}$

Short Answer

Expert verified

The differentiation for the series

$f (x) = \sum_{n = 0}^{\infty} a_{n} x^{n} f (x) = \sum_{n = 0}^{\infty} (a_{n} x^{n = 1})$

The given series is

$f (x) = \sin x = \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k}}{(2 k + 1)!} x^{2 k + 1}$

The differentiation of the above will be

$f (x) = \sum_{k = 0}^{\infty} (2 k + 1) (\frac{{(- 1)}^{k}}{(2 k + 1)!} x^{2 k})$

Step by step solution

differentiation of given power series

The differentiation for the series

$f (x) = \sum_{n = 0}^{\infty} a_{n} x^{n} f (x) = \sum_{n = 0}^{\infty} (a_{n} x^{n = 1})$

The given series is

$f (x) = \sin x = \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k}}{(2 k + 1)!} x^{2 k + 1}$

The differentiation of the above will be

$f (x) = \sum_{k = 0}^{\infty} (2 k + 1) (\frac{{(- 1)}^{k}}{(2 k + 1)!} x^{2 k})$

Final proof

$f (x) = \sum_{k = 0}^{\infty} (2 k + 1) (\frac{{(- 1)}^{k}}{(2 k + 1)!} x^{2 k})$

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Question 18: In Problems, find a power series expansion for $f (X)$ , given the expansion for f(x).
role="math" localid="1664283848012" $f (x) = \sin x = \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k}}{(2 k + 1)!} x^{2 k + 1}$

Short Answer

Step by step solution

differentiation of given power series

Final proof

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Question 18: In Problems, find a power series expansion for f(X) , given the expansion for f(x). role="math" localid="1664283848012" fx=sinx=∑k=0∞(-1)k(2k+1)!x2k+1

Short Answer

Step by step solution

differentiation of given power series

Final proof

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Geometry

Pure Maths

Decision Maths

Mechanics Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Question 18: In Problems, find a power series expansion for $f (X)$ , given the expansion for f(x).
role="math" localid="1664283848012" $f (x) = \sin x = \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k}}{(2 k + 1)!} x^{2 k + 1}$