# How to show the convergence of this infinite series: $\frac{x}{1+x}- \frac{x^2}{1+x^2}+ \frac{x^3}{1+x^3}\dots$

My series is $$\frac{x}{1+x}-\frac{x^2}{1+x^2}+ \frac{x^3}{1+x^3}-.....$$

Given: $0<x<1$

I see that my nth term is $(-1)^{n+1} (\frac{x^n}{1+x^n})$

My approach was to use Dirichlet's test. I see that $\frac1{1+x^n}$ is a monotone decreasing sequence converging to $0$.

So, I need to show that the partial sum sequences of the series $(-1)^{n+1} (x^n)$ is bounded. Is it true?

If not, how do I approach this problem?

• I'm sorry, is the general term $\frac{x^k}{1+x^k}$? – Alex Aug 26 '14 at 15:29
• Aren't you working with another alternating series again? – Quang Hoang Aug 26 '14 at 15:31
• Yes, I've been working with a lot of infinite series, lol. I got the last one, I'm stuck with this one. – Diya Aug 26 '14 at 15:32
• Learn to use LaTeX – Shahar Aug 26 '14 at 15:33
• Just show the terms are going to zero and apply the alternating series test. – Ian Mateus Aug 26 '14 at 15:33

Your infinite series is absolutely convergent: $$\left| \frac{x}{1+x}-\frac{x^2}{1+x^2}+ \frac{x^3}{1+x^3}-....\right|\leq\sum_{k=1}^{\infty}\left|\frac{x^k}{1+x^k}\right|\leq\sum_{k=1}^{\infty}x^k=\frac{x}{1-x}, \quad 0<x<1.$$