Math 2300 Section 005 – Calculus II – Fall 2010
Review on Series Convergence – October 6, 2010
§ Summary of Convergence of Series
§ Some General Notes

Pulling a finite number of terms off from a series will not affect convergence / divergence.

You can pull series apart like integrals (in terms of sums/difference and multiplication by a constant).

Series notation is shorthand, and in most cases series can be rewritten in any number of different ways.
§ Limit of Series Terms (Theorem 9.2 Part (c))

The series diverges if or if does not exist.

Conversely, just knowing doesn't tell us anything about convergence (except in the case of the Alternating Series Test, but there we also have to know that the magnitude of the terms is always getting smaller and the series is alternating).
§ Geometric Series

For finite geometric series .

For an infinite geometric series provided that .
§ series

The series converges if and diverges if .
§ Integral Test

Let be a positive integer and for positive integers .

converges if converges.

diverges if diverges.
§ Comparison Test

Consider two series and with .

If diverges, then diverges.

If converges, then converges.
§ Limit Comparison Test

Consider two series and with , and where .

If one of the series diverges, then the other does as well.

If one of the series converges, then the other does as well.
§ Ratio Test

The series converges (absolutely) if .

The series diverges if .

The series may converge or may diverge if .
§ Alternating Series Test

The alternating series converges if and .