Series Slides A review of convergence tests Roxanne M. Byrne University of Colorado at Denver

N th Term Test This test can be applied to any series

N th Term Test You must evaluate: Lim a n n   Where { a n } is the sequence of terms of the series

N th Term Test Conclusion: n   If lim a n  0, the series diverges If lim a n = 0, the test fails n   Where { a n } is the sequence of terms of the series

N th Term Test Remarks: Remember, if the limit is zero, THE TEST FAILS. This means you must try a different test. Sometimes the limit is not easy to evaluate. In this case, try other test that you think might be more productive first. Conversely, some of the other tests need this limit evaluated also. Remember this test if the limit is not zero.

Integral Test This test can be applied only to positive term series

INTEGRAL TEST You must: Find a continuous function, f(x), such that f(n) = a n Verify that f(x) is a decreasing function Determine if  f(x) dx converges Where { a n } is the sequence of terms of the series

INTEGRAL TEST Conclusion: If the integral converges then the series converges If the integral diverges then the series diverges

INTEGRAL TEST Remarks: This is both a convergence and divergence test If f(x) is an increasing function, go to the N th Term Test. This test requires that the function can be integrated. It will not work for series whose terms have factorials in them.

Comparison Test This test can be applied only to positive term series

COMPARISON TEST You must: Decide if you think the series converges or diverges If you think it converges, you must find a larger termed series that you know converges. If you think it diverges, you must find a smaller positive termed series that you know diverges

COMPARISON TEST Conclusion: If you find a larger termed convergent series, then your series converges. If you find a smaller positive termed divergent series, then your series diverges. If you cannot find an appropriate comparison series, the test fails.

COMPARISON TEST Remarks: As with the N th Term Test, when the test fails, it means you must try another test. The test works well with series that look almost like a geometric series or a p-series. The major disadvantages of this test: 6You must decide beforehand if the series converges or diverges. 6You must find a corresponding comparison series

Limit Comparison Test This test can be applied only to positive term series

LIMIT COMPARISON TEST You must: Decide if you think the series converges or diverges If you think it converges, you must find a positive termed convergent series that has the same end behavior as yours. If you think it diverges, you must find a positive termed divergent series that has the same end behavior as yours. Evaluate where a n and b n are the terms of your two series

If 0 < < , then both series converge or both series diverge. If equals zero or increases without bound or does not exist, then test fails. LIMIT COMPARISON TEST Conclusion:

LIMIT COMPARISON TEST When the test fails, you must either find another comparison series or you must try another test. The test works well with series that look almost like a geometric series or p-series. The major disadvantages of this test: 6You must decide beforehand if the series converges or diverges. 6You must find a corresponding comparison series Remarks:

Ratio Test This test can be applied only to positive term series

You must: Evaluate u n + 1 Evaluate the ratio Evaluate lim RATIO TEST n   Where { u n } is the sequence of terms

RATIO TEST Conclusion: If the limit < 1 then the series converges If the limit > 1 then the series diverges If the limit = 1 then the test fails

Remarks: This is both a convergence and divergence test This test can be used to prove absolute convergence This test will not work on series whose terms are rational functions of n. For these series, use the Limit Comparison Test and the end behavior of the terms. RATIO TEST This test works well with series whose terms have factorials in them.

The N th Root Test This test can be applied only to positive term series

THE N TH ROOT TEST You must: Find Evaluate Where { a n } is the sequence of terms of the series

THE N TH ROOT TEST Conclusion: If the limit < 1 then the series converges If the limit > 1 then the series diverges If the limit = 1 then the test fails

THE N TH ROOT TEST Remarks: This is both a convergence and divergence test This test can be used to prove absolute convergence This test will not work on series whose terms are rational functions of n. For these series, use the Limit Comparison Test and the end behavior of the terms. This test works well with series whose terms have powers of n in them. This test does not work well with series whose terms have factorials in them.

Absolute Convergence Test This test is used on series with varying signed terms

ABSOLUTE CONVERGENCE TEST You must: Let b n be the absolute value of the sequence of terms of your series Determine if the sum of b n is a convergent series by one of the positive term convergence tests.

Conclusion: If the sum of b n converges then the original series converges absolutely If the sum of b n converges then the original series converges conditionally or it diverges. ABSOLUTE CONVERGENCE TEST

Remarks: If the sum of b n diverges then you usually use the alternating series test to determine if the original series converges. If you want to determine the type of convergence of an alternating series, you would use this test first. ABSOLUTE CONVERGENCE TEST

Alternating Series Test This test can be applied only to series that have alternating terms

You must: Make sure the terms are alternating Define a new sequence, u n, as the absolute value of the terms of your sequence of terms. Prove that u n is a decreasing sequence. Evaluate lim u n ALTERNATING SERIES TEST n  

ALTERNATING SERIES TEST Conclusion: If the limit is zero, then alternating series converges.

ALTERNATING SERIES TEST If you need to determine if the series is absolutely or conditionally convergent, you must test to see if  u n converges using a positive term series test. If the lim u n  0 or if u n is an increasing sequence, use the N th Term Test. Remarks:

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