2 Definition of Convergence for an infinite series: Let be an infinite series of positive terms.The series converges if and only if the sequence of partial sums,, converges. This means:Divergence Test:If , the series diverges.Example: The seriesis divergent sinceHowever,does not imply convergence!
7 11.3 11.4 11.5 11.6 The Integral Test The Comparison Tests Alternating Series11.6Ratio and Root Tests
8 Integral test:If f is a continuous, positive, decreasing function on withthen the series converges if and only if the improper integral converges.Example: Try the series: Note: in general for a series of the form:
9 Comparison test:If the series and are two series with positive terms, then:If is convergent and for all n, then converges.If is divergent and for all n, then diverges.(smaller than convergent is convergent)(bigger than divergent is divergent)Examples: which is a divergent harmonic series. Since the original series is larger by comparison, it is divergent.which is a convergent p-series. Since theoriginal series is smaller by comparison, it isconvergent.
10 Limit Comparison test: If the the series and are two series with positive terms, and ifwhere then either both series converge or both series diverge.Useful trick: To obtain a series for comparison, omit lower order terms in the numerator and the denominator and then simplify.Examples: For the series compare to which is a convergent p-series.For the series compare to which is a divergent geometric series.
11 Alternating Series test: If the alternating series satisfies:and then the series converges.Definition: Absolute convergence means that the series converges without alternating (all signs and terms are positive).Example: The series is convergent but not absolutely convergent.Alternating p-series converges for p > 0. Example: The series and the Alternating Harmonic series are convergent.
12 Ratio test: If then the series converges; If the series diverges. Otherwise, you must use a different test for convergence.If this limit is 1, the test is inconclusive and a different test is required.Specifically, the Ratio Test does not work for p-series.Example:
13 Strategy for Testing Series 11.7Strategy for Testing Series
14 Summary: Apply the following steps when testing for convergence: Does the nth term approach zero as n approaches infinity? If not, the Divergence Test implies the series diverges.Is the series one of the special types - geometric, telescoping, p-series, alternating series?Can the integral test, ratio test, or root test be applied?Can the series be compared in a useful way to one of the special types?
16 ExampleDetermine whether the series converges or diverges using the Integral Test.Solution: Integral Test:Since this improper integral is divergent, the series (ln n)/n is also divergent by the Integral Test.