Power Sum Expansion and Convergence Questions

[MathematicalPhysicist]

1) develop the function f(x)=(e^x)sin(x) into a power sum over the point 0.
2) find the convergence radius R of $$\sum_{\substack{0<=n<\infty}}\frac{(n!)^2}{(2n)!}x^n$$ and say if it converges or diverges at x=-R, x=R.

about the second question i got that R=4, through hadamard test, but i didnt succeed in asserting if at x=R it diverges or converges, at x=-R i think it converges because it's an alternating sign sum, and according to leibnitz theorem it does.

about the first question here what i got:
i needed to find an equation for the derivative of $$f^{(n)}(x)$$, here what i got:
$$f^{(n)}(x)=(g(x)h(x))^{(n)}=\binom{n}{n}g^{(n)}(x)h(x)+\binom{n}{n-1}g^{(n-1)}(x)h'(x)+...+\binom{n}{n-1}g'(x)h^{(n-1)}(x)+\binom{n}{n}g(x)h^{(n)}(x)$$ which i employed at the function which i got, is this equation correct?

thanks in advance.

 

[eok20]

for the first question, it is probably easiest to find the power series for e^x and for sin(x) about 0 and then multiply them together.

 

[MathematicalPhysicist]

i thought about it, but i wasn't sure, it would be accaptable.
but on a second thought it does make a perfect sense.

what about my second question?
p.s
about my first question, how do i represent the product of the sums of e^x and sin(x) as one sum?

 

[StatusX]

You could use the fact that:

$$\sin x = \frac{1}{2i} (e^{ix}-e^{-ix})$$

so we can write:

$$e^x \sin x = \frac{1}{2i}(e^{(1+i)x}-e^{(1-i)x})$$

Then, for example,

$$e^{(1+i)x} = 1+ (1+i)x+ \frac{1}{2}(1+i)^2 x^2+...$$

To compute powers of $1 \pm i$, it is probably easiest to rewrite it as $r e^{i\theta}$ for an appropriate choice of $r$ and $\theta$.

 

[benorin]

$$e^{x}\sin{x} = \left( \sum_{n=0}^{\infty} \frac{x^{n}}{n!}\right) \left( \sum_{m=0}^{\infty} \frac{x^{2m+1}}{(2m+1)!}\right) = \sum_{n=0}^{\infty}\sum_{k=0}^{n} \frac{x^{n-k}}{(n-k)!} \frac{x^{2k+1}}{(2k+1)!} = \sum_{n=0}^{\infty}\sum_{k=0}^{n} \frac{x^{n+k+1}}{(n-k)!(2k+1)!}$$

 

[MathematicalPhysicist]

can someone help on the other question, does it converge or diverge at x=4, and how to prove it?

thanks.

 

[benorin]

Well the product of to absolutely convergent series is absolutely convergent, and the series used converge for all $$-\infty < x<\infty$$: so, yes, it does converge at x=4.

 

[MathematicalPhysicist]

but R doesn't equal $$\infty$$, i know that for every |x|<R the sum converges but here i need to find what happens when x=R.

 

[StatusX]

Find the ratio of successive terms at x=4. Do they get bigger or smaller?

 

[MathematicalPhysicist]

you mean, to use d'almbert test, ok, thanks.