Originally Posted by

**emjones** Question:

A)

$\displaystyle s(x) = \sum\frac{x^{2n+1}}{(2n+1)!}$

and $\displaystyle c(x) = \sum\frac{x^{2n}}{2n!}$

Find the radii of convergence of these series and show that $\displaystyle s'(x) =c(x)$ and $\displaystyle c'(x) = s(x)$ for all $\displaystyle x\in\mathbb{R}$

B)

For every $\displaystyle \alpha\in\mathbb{R}$, let $\displaystyle h\alpha(x)$ denote the function:

$\displaystyle s(x+\alpha)c(\alpha-x) + s(\alpha-x)c(x+\alpha)$

show that $\displaystyle h\alpha(x)$ has constant value $\displaystyle s(2\alpha)$

C)

Use this result or by another means to show that for all $\displaystyle \beta,\gamma\in\mathbb{R}$

$\displaystyle s(\beta+\gamma)=s(\beta)c(\gamma)+s(\gamma)c(\beta )$

Solution:

It is difficult for me to write it out on here because I don't know how to write all the maths properly like everybody else on maths forum because I am new!

but for A) I think I must be going wrong because for both $\displaystyle s(x)$ and $\displaystyle c(x)$ the ratio test is telling me they are both null series and thus they converge for all $\displaystyle x$. Is this correct?

b) I am unsure how to begin this part, maybe i need to use part a)

any advice would be good as I'm having trouble even really starting this question!

Thank you.