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Q-18E

Expert-verifiedFound in: Page 434

Book edition
9th

Author(s)
R. Kent Nagle, Edward B. Saff, Arthur David Snider

Pages
616 pages

ISBN
9780321977069

**Question 18:** **In Problems, find a power series expansion for** $f\left(X\right)$ **, given the expansion for f(x).**

role="math" localid="1664283848012" $f\left(x\right)=\mathrm{sin}x=\sum _{k=0}^{\infty}\frac{{(-1)}^{k}}{(2k+1)!}{x}^{2k+1}$

The differentiation for the series

$f\left(x\right)=\sum _{n=0}^{\infty}{a}_{n}{x}^{n}f\left(x\right)=\sum _{n=0}^{\infty}\left({a}_{n}{x}^{n=1}\right)$

The given series is

$f\left(x\right)=\mathrm{sin}x=\sum _{k=0}^{\infty}\frac{{(-1)}^{k}}{(2k+1)!}{x}^{2k+1}$

The differentiation of the above will be

$f\left(x\right)=\sum _{k=0}^{\infty}(2k+1)\left(\frac{{(-1)}^{k}}{(2k+1)!}{x}^{2k}\right)$

$f\left(x\right)=\sum _{k=0}^{\infty}(2k+1)\left(\frac{{(-1)}^{k}}{(2k+1)!}{x}^{2k}\right)$

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