Does $\sum^{\infty}_{1}\frac{1}{k+1} = \sum^{\infty}_{2}\frac{1}{k}$ diverge because $\sum^{\infty}_{1}\frac{1}{k}$ diverges? If you manipulate the index of a series does it still converge/diverge?
For example:
Does $\sum^{\infty}_{1}\frac{1}{k+1} = \sum^{\infty}_{2}\frac{1}{k}$ diverge because $\sum^{\infty}_{1}\frac{1}{k}$ diverges?

Changing a finite number of terms will not alter whether it converges or not. For instance, $$ \sum_{n=1}^\infty \frac{1}{n^2} \text{ and } \sum_{n=2}^\infty \frac{1}{n^2} $$
both converge, just to a different value.
However, be careful. Changing the index can create poles causing the sum to be undefined. Take for example,
$$ \sum_{n=0}^\infty \frac{1}{n^2} $$
The whole sum is undefined, because $\frac{1}{0}$ is a term.
