if (c(x)=sum_{n = 0}^{infty}\frac{2n}{4n^{2}+1}x^{2n}) and (s(x)=sum_{n = 0}^{infty}\frac{2n +…

if (c(x)=sum_{n = 0}^{infty}\frac{2n}{4n^{2}+1}x^{2n}) and (s(x)=sum_{n = 0}^{infty}\frac{2n + 1}{(2n+1)^{2}+1}x^{2n + 1}), find the power series of (c(x)-s(x)).\n(sum_{n = 0}^{infty})\nsubmit question jump to answer

if (c(x)=sum_{n = 0}^{infty}\frac{2n}{4n^{2}+1}x^{2n}) and (s(x)=sum_{n = 0}^{infty}\frac{2n + 1}{(2n+1)^{2}+1}x^{2n + 1}), find the power series of (c(x)-s(x)).\n(sum_{n = 0}^{infty})\nsubmit question jump to answer

Answer

Explanation:

Step1: Write out the subtraction

[C(x)-S(x)=\sum_{n = 0}^{\infty}\frac{2n}{4n^{2}+1}x^{2n}-\sum_{n = 0}^{\infty}\frac{2n + 1}{(2n+1)^{2}+1}x^{2n + 1}] We can combine these two series into a single series (\sum_{k=0}^{\infty}a_{k}x^{k}). When (k = 2n) (even), (a_{2n}=\frac{2n}{4n^{2}+1}), and when (k=2n + 1) (odd), (a_{2n+1}=-\frac{2n + 1}{(2n+1)^{2}+1}). So the power - series of (C(x)-S(x)) is (\sum_{n = 0}^{\infty}\left(\frac{2n}{4n^{2}+1}x^{2n}-\frac{2n + 1}{(2n+1)^{2}+1}x^{2n + 1}\right))

Answer:

(\sum_{n = 0}^{\infty}\left(\frac{2n}{4n^{2}+1}x^{2n}-\frac{2n + 1}{(2n+1)^{2}+1}x^{2n + 1}\right))