Infinite Series: A Synopsis

Famous Series

Geometric Series • if converges with sum • if diverges P-Series • for diverges Harmonic Series • diverges • this is a special case of the P-Series for P=1

Alternating Harmonic Series • converges to ln(1+1) = ln(2) using series for ln(1+x) below. Exponential • converges  where Sin • converges Cos • converges ln (1+x)  • 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;

arctan (x)  • you can arrive at this relation by integrating a Geometric Series in -t^2 term-by-term.     Useful Inequalities

Always

1. 2. 3. 4. 5. 6. Eventually

1. 2. for any 3. (!) Remember this when using the Test for Divergence

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 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.

-- DickFurnas - 21 Oct 2008
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Topic revision: r4 - 2008-10-29 - DickFurnas   Edit Attach Copyright © by the contributing authors. All material on this collaboration platform is the property of the contributing authors.