1. ## Series Convergence

What are the values of k that make the series convergent?

2. ## Re: Series Convergence

Just a thought, give this a try: Let $\displaystyle u = 2 + e^{2x}$. Then we have that $\displaystyle e^x = \sqrt{u - 2}$.

$\displaystyle \sum_{k = 1}^{\infty} \dfrac{\sqrt{u - 2}}{u^k}$

which looks a lot simpler.

-Dan

3. ## Re: Series Convergence

Originally Posted by topsquark
Just a thought, give this a try: Let $\displaystyle u = 2 + e^{2x}$. Then we have that $\displaystyle e^x = \sqrt{u - 2}$.

$\displaystyle \sum_{k = 1}^{\infty} \dfrac{\sqrt{u - 2}}{u^k}$

which looks a lot simpler.

-Dan

I want x to go from 1 to infinite, not k. I want to figure what are the values of k.
For example, when k = 0, the series diverges, but k = 1, it converges.

4. ## Re: Series Convergence

by plugging random values of k, I observed that the sum increases very very slightly like it will reach finite value for k > 0.5.

If k <= 0.5, the sum will increase gradually to infinite.

Does that mean the series is converges when k > 0.5?
How to prove that mathematically?

5. ## Re: Series Convergence

$\displaystyle 0<\frac{e^x}{\left(2+e^{2x}\right)^k}<\frac{e^x}{e ^{2k x}}=\left(e^{1-2k}\right)^x$

6. ## Re: Series Convergence

thanks Idea and topsquark.