…an overview of sections 9.3 – 9.4

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: – p-test Comparison Test Alternating Series Test Ratio Test Nth Root Test

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 510 # 5

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 519 #12

Alternating Series Test If – and the Series converges Pg 519 #26

Ratio and Root Tests Consider the series let if: –  < 1 series converges –  > 1 series diverges –  = 1 ??????????? Example: Pg 519 #15

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

Summary – Series Check List Always try comparison test first! Yes! - done No? Expression has factorials – ratio test Positive, monotonic decreasing? – Integral test Expression has powers – root test Alternating terms – alternating series test some samples…