a+ar+ar^2 + ... The term of the series is ar^n. This series converges if |r|<1 and diverges otherwise. If it converges, it converges to a/(1-r).
The term of the p-series is 1/n^p. This series converges if p>1 and diverges otherwise. If p=1, this is the harmonic series.
This kind of series cancels itself out. For instance if you write out the sum over n from 1 to infinity of (1/n-1/(n+1)), you'll get 1/1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + ..., and all of the terms except the "1" in front will cancel each other out, so the series will converge to 1.
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