Mathematical Methods in the Physical Sciences/Chapter 1/Section 4

Overview

Infinite series are either convergent (have a sum) or divergent (don't have a sum).
Playing with the sums of divergent series leads to problems. Don't do it.
Sometimes a series of positive terms is divergent, but the corresponding alternating system is not.
The harmonic series is <math>1 + {1 \over 2} + {1 \over 3} + ... + {1 \over n} + ...</math> and is divergent.
The alternating harmonic series is convergent, but can be made to have any sum by rearranging the terms!
The partial sum of any series is:
<math>S_n = a_1 + a_2 + a_3 + ... + a_n</math>
The sum of a series is:
<math>S = \lim_{n \to \infty} S_n</math>
If the sum does not exist, the series is divergent. If it does, it is convergent.
The remainder of a convergent series is:
<math>R_n = S - S_N</math>
For convergent series, the remainder tends to 0 as n approached infinity.
Divergent series have no sum and thus no remainder.