Harrell's little crib sheet on series

© Copyright 2000 by Evans M. Harrell II

Thou shalt know the following series:

  1. The geometric series: convergent for |x| < 1 (variants)
  2. The arithmetic series: divergent even though a0 tends to zero. The arithmetic series is actually the borderline case for the p-series, which converges if p > 1. However, there is no simple formula for the sum of the general p-series.
  3. The Taylor series for These converge for all x.

Fiddling with series


If series converge you can fiddle up other convergent series if you: (See some examples.)

Spotting elephants


In the Infinite Seriesian Jungle, there are only two kinds of animals, elephants (divergent series) and bugs (convergent series). There are two basic ways to recognize elephants, i.e., to show that series diverge. Other tests for divergence are really special cases of the comparison test.
  1. If the parts of the beast aren't small, it's an elephant. (If an does not tend to 0, the series does not converge.)
  2. (comparison test) If a beast in the jungle is bigger than an elephant, it's another elephant!
Just as there are two kinds of elephants, there are two ways a series can fail to converge. The partial sums may tend to infinity, but another kind of divergence ("by oscillation") happens if they simply wander around finitely without ever committing themselves (examples.)

Squashing series down


There are several ways to squash bugs in the jungle, i.e., to show that a series converges. Here we mention only a few of them. Most tests for convergence use the absolute values of an, because if the absolute value series converges, then so does the series where an can be positive or negative.
  1. (comparison test) If a beast in the jungle is smaller (in absolute value) than a bug, it is another bug!
  2. (alternating series test) If it has little parts and they wiggle back and forth, it's a bug: If and the signs alternate (strictly, at least after the first million), then the series converges.
  3. (Taylor's formula with remainder) A series can be proved to converge if you can recognize it as a convergent Taylor series with some particular value of x.
(examples)

Hunting for elephants or bugs


Some tests go both ways, but these are really special cases of the comparison tests:
  1. (ratio test) If then the series converges. If > 1, it diverges. If = 1, all bets are off.
  2. (limit comparison test) If bn > 0 and no matter how large L is, then the a series and the b series either both converge or both diverge.
  3. (integral test) If a(x) is a positive function, then the series with an converges or diverges together with the integral of a(x) from any finite value to .
(examples)

Some tips

  1. If you are only asked about convergence, the first million or so terms don't matter at all.
  2. A Taylor or power series becomes an ordinary series as soon as you fix x. Usually, convergence of these series comes from either Lagrange's formula or the ratio test.
  3. By its very nature, an error estimate is hardly ever exact. So, simplify, simplify! Identify something bigger to replace something ugly (like 1 in favor of cos(exp(x1/2)).