…an overview of sections 11.2 – 11.6 Testing Convergence …an overview of sections 11.2 – 11.6

Some Fundamentals… For a series to converge the elements of the series MUST converge to zero! but This is a necessary but not sufficient condition! Example: does the following (harmonic series) converge?

A few major methods… Integral Test: (11.3) Comparison Test (11.4) p-test Comparison Test (11.4) Alternating Series Test (11.5) Ratio Test (11.6) Nth Root Test (11.6)

Integral Test Applies to monotonic, positive, decreasing functions Use the connection between summation and integration Express generating function for series as an integrand: Example: does converge? Compare this to Series converges if the integral does! Pg 704 #27

Comparison Test Sorta “common sense”: “if series A converges and all of series B terms are less than or equal to series A terms then series B also converges” The “catch” (there is always a catch!): the terms must be non-negative. Example: Test convergence (or divergence) of: A) B) Pg 709 #27

Alternating Series Test If and the Series converges Pg 713 #13

Ratio and Root Tests Consider the series let if: Example: r < 1 series converges r > 1 series diverges r = 1 ??????????? Example:

Ratio and Root Tests Consider the series let if: Example: r < 1 series converges r > 1 series diverges r = 1 ??????????? Example: