Convergence or Divergence of Infinite Series

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 series is divergent since However, does not imply convergence!

Special Series

Geometric Series: The Geometric Series: converges for If the series converges, the sum of the series is: Example: The series with and converges . The sum of the series is 35.  

p-Series: The Series: (called a p-series) converges for and diverges for Example: The series is convergent. The series is divergent.  

Convergence Tests

11.3 11.4 11.5 11.6 The Integral Test The Comparison Tests Alternating Series 11.6 Ratio and Root Tests

Integral test: If f is a continuous, positive, decreasing function on with then the series converges if and only if the improper integral converges. Example: Try the series: Note: in general for a series of the form:

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 the original series is smaller by comparison, it is convergent.

Limit Comparison test: If the the series and are two series with positive terms, and if where 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.

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.

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:

Strategy for Testing Series 11.7 Strategy for Testing Series

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?

Summary of all tests:

Example Determine 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.

More examples

Example 1 Since as an  ≠ as n  , we should use the Test for Divergence.

Example 2: Using the limit comparison test Since an is an algebraic function of n, we compare the given series with a p-series. The comparison series for the Limit Comparison Test is where

Example 3 Since the series is alternating, we use the Alternating Series Test.

Example 4 Since the series involves k!, we use the Ratio Test.

Example 5 Since the series is closely related to the geometric series , we use the Comparison Test.