Strategy for Testing Series
STRATEGY FOR TESTING SERIES 1. Check for known series. p-series converges if . diverges if . (Note: When , the series is the harmonic series.)
geometric series converges if . diverges if .
telescoping series converges if a real number. diverges otherwise.
2. Use a test.
NOTE: When testing a series for convergence or divergence, two components must be shown: (i) State the test that is used: “Therefore, the series [converges/diverges] by the [name of test].”
(ii) Demonstrate that the conditions of the test are met.
Test for Divergence diverges if .
NOTE: The test is inconclusive if . Try another test.
Basic Comparison Test converges if and converges. diverges if and diverges.
NOTE: If the inequality goes the wrong way for the conclusion, use the L.C.T below.
Limit Comparison Test converges if and are positive, , and converges. diverges if and are positive, , and diverges.
Both comparison tests are useful for rational and algebraic functions of n.
Integral Test converges if converges such that . diverges if diverges such that .
NOTE: It must be shown that is continuous, positive, and decreasing on .
The integral and comparison tests cannot be used on a series with both positive and negative terms; however, they may be applied to to test for absolute convergence. In this case, if the tests indicate divergence, it cannot be concluded that diverges, only that it does not converge absolutely.
Ratio Test converges if . diverges if .
NOTE: The test is inconclusive if . Try another test. NOTE: This test is useful for factorial and exponential functions of n.
Root Test converges if . diverges if .
NOTE: The test is inconclusive if . Try another test. NOTE: This test is useful for exponential functions of n.
Unlike the integral and comparison tests, the ratio and root tests do determine convergence or divergence of the original series , even if the series has both
geometric series converges if . diverges if .
telescoping series converges if a real number. diverges otherwise.
2. Use a test.
NOTE: When testing a series for convergence or divergence, two components must be shown: (i) State the test that is used: “Therefore, the series [converges/diverges] by the [name of test].”
(ii) Demonstrate that the conditions of the test are met.
Test for Divergence diverges if .
NOTE: The test is inconclusive if . Try another test.
Basic Comparison Test converges if and converges. diverges if and diverges.
NOTE: If the inequality goes the wrong way for the conclusion, use the L.C.T below.
Limit Comparison Test converges if and are positive, , and converges. diverges if and are positive, , and diverges.
Both comparison tests are useful for rational and algebraic functions of n.
Integral Test converges if converges such that . diverges if diverges such that .
NOTE: It must be shown that is continuous, positive, and decreasing on .
The integral and comparison tests cannot be used on a series with both positive and negative terms; however, they may be applied to to test for absolute convergence. In this case, if the tests indicate divergence, it cannot be concluded that diverges, only that it does not converge absolutely.
Ratio Test converges if . diverges if .
NOTE: The test is inconclusive if . Try another test. NOTE: This test is useful for factorial and exponential functions of n.
Root Test converges if . diverges if .
NOTE: The test is inconclusive if . Try another test. NOTE: This test is useful for exponential functions of n.
Unlike the integral and comparison tests, the ratio and root tests do determine convergence or divergence of the original series , even if the series has both