# Thread: Help solving taylor series.

1. ## Help solving taylor series.

I've just started working with Taylor series and can't figure out how to find the series of the following function:

g(x)= $\int_0^x [2e^v+3e^{-v}] dv$

Could anyone please show me the solution?

2. Originally Posted by goblinf
I've just started working with Taylor series and can't figure out how to find the series of the following function:

g(x)= $\int_0^x [2e^v+3e^{-v}] dv$

Could anyone please show me the solution?

$\int\limits_0^x [2e^v+3e^{-v}] dv=2e^x-2-3e^{-x}+3=2e^x-3e^{-x}+1$ . Now just remember that

$\displaystyle{e^x=\sum\limits^\infty_{k=0}\frac{x^ k}{k!}\,,\,\,\forall x\in\mathbb{R}}$

Tonio

3. And for which values of x does the series converge?

4. Originally Posted by goblinf
And for which values of x does the series converge?
Tonio stated that after the series, surely? It holds for all $x$.

5. Originally Posted by TheCoffeeMachine
Tonio stated that after the series, surely? It holds for all $x$.
Could you please describe this a bit more. How do you see that it holds for all $x$?

6. It's well known that the Taylor series for $\displaystyle e^x$ is convergent for all $\displaystyle x$.

But this is also relatively easy to show using the ratio test.