Convergence Tests

Applying limits may allow us to determine the finite sum of infinitely many terms.

• By the end of the section, you should be able to do the following:

  • Determine whether a series converges or diverges.

  • Approximate the sum of a series.

  • Determine the radius of convergence and interval of convergence for a power series.

• By the end of the section, you should know the following:

  • The nth term test is a test for divergence of a series.

  • The integral test is a method to determine whether a series converges or diverges.

  • In addition to geometric series, common series of numbers include the harmonic series, the alternating harmonic series, and p-series.

  • The comparison test is a method to determine whether a series converges or diverges.

  • The limit comparison test is a method to determine whether a series converges or diverges.

  • The ratio test is a method to determine whether a series of numbers converges or diverges.

  • The alternating series test is a method to determine whether an alternating series converges.

  • If an alternating series converges by the alternating series test, then the alternating series error bound can be used to bound how far a partial sum is from the value of the infinite series.

  • A series may be absolutely convergent, conditionally convergent, or divergent.

  • If a series converges absolutely, then it converges.

  • If a series converges absolutely, then any series obtained from it by regrouping or rearranging the terms has the same value.

  • If a power series converges, it either converges at a single point or has an interval of convergence.

  • The ratio test can be used to determine the radius of convergence of a power series.

  • The radius of convergence of a power series can be used to identify an open interval on which the series converges, but it is necessary to test both endpoints of the interval to determine the interval of convergence.

  • If a power series has a positive radius of convergence, then the power series is the Taylor series of the function to which it converges over the open interval.

  • The radius of convergence of a power series obtained by term-by-term differentiation or term-by-term integration is the same as the radius of convergence of the original power series.