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---+ !!Infinite Series: A Synopsis
%BEGINOVERVIEW%

%ENDOVERVIEW%

---+ Famous Series
%INCLUDE{FamousSeries}%

---+ [[Useful Limits]]
%INCLUDE{UsefulLimits}%

---+ Useful Inequalities
%INCLUDE{UsefulInequalities}%

---+ New Series From Old
%INCLUDE{NewSeriesFromOld}%

---+ [[Convergence Tests]]
%INCLUDE{ConvergenceTests]]%

---+ Tips for Series
   * 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 [[UsefulLimits]] 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" <latex>\frac{n^3}{n^4}</latex> as <latex>n \rightarrow \infty</latex> then the series "looks like" <latex>\frac{1}{n}</latex> 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 <latex>\ln{n} < n^k </latex> for any <latex>k > 0 </latex> 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, <latex>\sum \frac{\ln{n}}{n^2}</latex> "looks like" <latex>\sum\frac{1}{n^2}</latex> so converges
      * more carefully, <latex>\ln{n} < \sqrt{n}</latex> (eventually),
        %BR%so<latex>\sum\frac{\ln{n}}{n^2} < \frac{\sqrt{n}}{n^2} = \sum\frac{1}{n^{\frac{3}{2}}}</latex>,
        %BR%which is a convergent p-series.

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-- Main.DickFurnas - 21 Oct 2008