Prove by Induction: $1- \frac{1}{2}+ \frac{1}{3}- \frac{1}{4} + \ldots + \frac{(-1)^n}{n}$ is always positive How do you prove by induction that $1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\ldots+\frac{(-1)^n}{n}$ is always positive.
The base case works out and the inductive step is n=k, so $1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\ldots+\frac{(-1)^k}{k}$ holds true for the statement. I am stuck on the n = k+1. 
if any one can give an idea, it would help alot.

Proving that $1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\ldots+\frac{(-1)^n}{n}$ is always positive is equivalent to proving that:
$1+\frac{1}{3}+\dots +\frac{1}{2n-1} > \frac{1}{2}+\frac{1}{4}+\dots +\frac{1}{2n}$
For all $n\geq 1$
Clearly this is true when $n=1$. Assuming this is true for $n=k$, how could you show this is true for $n=k+1$?
