Originally Posted by

**ScottO** The harmonic series:

$\displaystyle \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + ...$

is divergent, i.e. sums to $\displaystyle \infty$.

This geometric series:

$\displaystyle \frac{1}{1} + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ...$

is convergent to 2.

In general, is there a way to know if removing all the elements of a convergent series, from a divergent series, if the result will be divergent or convergent?

It makes sense to me, that if I were to remove $\displaystyle \frac{1}{1}$ from the harmonic series, it would remain divergent. $\displaystyle \infty - 1$ is still $\displaystyle \infty$ after all. But how much can I take away from something infinite, and have it remain so? Thinking the other way, how much do I have to remove to get it to converge?

-Scott

P.S.

Is Calculus the right forum for questions on series and sequences?