Educator

Series

Main definitions and theorem:

Definitions:

• The notation stands for the series a₀ + a₁ + a₂ + …

• The sequence of partial sums of the series is the sequence

  

• The phrase the series ∑a_n converges (to a limit L) means that the sequence of partial sums {s_n} converges (to L).

• The same definition holds for diverges, diverges to ∞, and diverges to −∞.

A geometric series is one where each term is the previous term multiplied by the same common ratio:

Theorem (Test For Divergence): If is a series, and the sequence {a_n} converges to something other than 0, or if it diverges, then the series diverges.

Hints and tips:

• Remember to distinguish between the sequence of terms {a_n}, the series ∑a_n, and the sequence of partial sums {s_n}. All three are different!

• In determining whether a series converges or diverges, you can ignore the first few terms. What is important is what the later terms do. However, the first few terms do matter in determining what limit a series converges to.

• Don’t try to analyze series by plugging numbers into a calculator. This is extremely unreliable.

• The Test For Divergence is usually the first and easiest test to check for series. However, remember that it is a one way test. If the sequence does converge to 0, then TFD tells you nothing about the series.

• You can sometimes use partial fractions to analyze a series involving rational functions. Then write out some partial sums of the series and see if the terms cancel. These are called telescoping series.

• It is sometimes useful to invoke algebraic identities such as ln a/b = ln a − ln b.

• There are several different versions of the formula for the sum of a geometric series, depending on whether the series starts at n = 0, n = 1, and so on. The easiest one to remember, which works in all situations, is

  	   first term   
  	1 − common ratio

  assuming the ratio has absolute value less than 1.