Infinite Series: A Synopsis

Table of Contents
Infinite Series: A Synopsis Famous Series Geometric Series P-Series Harmonic Series Alternating Harmonic Series Exponential Sin Cos ln (1+x) arctan (x) Useful Inequalities Always Eventually Tips for Series

Famous Series

where

you can arrive at this relation by integrating a Geometric Series in -t term-by-term.

if x=-1 , i.e. ln(1+(-1)) = ln(0) , this is the negative of the Harmonic Series which diverges toward -&infty;

you can arrive at this relation by integrating a Geometric Series in -t^2 term-by-term.

Often the hardest part of showing convergence or divergence of a series is the indecision: What do I believe it does? After all, you'll have a tough time showing a series converges if it doesn't!
The limits listed in another section can help a lot with the Test for Divergence. Together with inequalities you can often get an idea of what to try to show. If the individual terms of the series "look like" as then the series "looks like" and you will want to show it diverges, perhaps even setting up a comparison, or limit comparison with 1/n itself.
Many limits boil down to "look like" ratios of polynomials after stripping out trig functions using the Useful Inequalities for trig functions.
The eventual behavior that for any leads to the peculiar rule of thumb that in lots of ratios ln(n) "looks like" 1 since any positive power of n will dominate it:
- informally, "looks like" so converges
- more carefully, (eventually),
  so,
  which is a convergent p-series.