How to determine Absolute Convergence and Conditional Convergence.

Question

In this hw problem it is asking to determine the absolute convergence values and place them in interval notation same as for conditional convergence. $$\sum_{n=1}^{\infty}\frac{x^n}{n}$$ My question is where would I go from this? (Hints would be preferred as I still need to learn how to approach these problems.)

Answer 1

Absolute convergence: $\lim\limits_{n\to\infty}|\frac{\frac{x^{n+1}}{n+1}}{\frac{x^n}{n}}|=|x|\lim\limits_{n\to\infty}\frac{n}{n+1}<1\,$ => $\,|x|<1$

Answer 2

If you haven't learned ratio test yet, then the only thing you have is geometric series. We need two facts: 1. a geometric series converges if the common ratio is between $-1$ and $1$ (open interval.) 2. The harmonic series diverges and the alternating harmonic series converges. Then we do cases:

If $\mid x \mid >1$ then the absolute version of your series is greater than $\sum 1/n$ and so diverges.

If $0\leq x <1$, then your series is less than $\sum x^n$ which is geometric and converges (absolutely.)

If $x=1$ then the series diverges (harmonic.)

If $x=-1$ then the series converges conditionally (alternating harmonic.)