IF Sec 11.6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS DEF:

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Presentation transcript:

IF Sec 11.6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS DEF: REMARK: IF Notice that if a series with positive terms, then absolute convergence is the same as convergence. Is called Absolutely convergent convergent Example: Example: Test the series for absolute convergence. Test the series for absolute convergence.

Is called conditionally convergent Sec 11.6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS Example: DEF: Test the series for absolute convergence. IF Is called Absolutely convergent convergent Example: DEF: Test the series for absolute convergence. Is called conditionally convergent if it is convergent but not absolutely convergent.

Sec 11.6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS Absolutely convergent convergent THM: Example: Determine whether the series converges or diverges.

Sec 11.6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS Absolutely convergent convergent THM:

Sec 11.6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS Series Tests Test for Divergence Integral Test Comparison Test Limit Comparison Test Alternating Test Ratio Test Let

Sec 11.6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS Ratio Test Let Example: Example: Test the series for absolute convergence. Test the series for absolute convergence.

Sec 11.6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS Ratio Test Let REMARK: Case L = 1 means that the test gives no information.

Sec 11.6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS TERM-101

Sec 11.6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS Series Tests Example: Test for Divergence Integral Test Comparison Test Limit Comparison Test Alternating Test Ratio Test Test the series for convergence. Root Test Let

Sec 11.6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS TERM-082

Sec 11.6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS REARRANGEMENTS Divergent If we rearrange the order of the terms in a finite sum, then of course the value of the sum remains unchanged. But this is not always the case for an infinite series. By a rearrangement of an infinite series we mean a series obtained by simply changing the order of the terms.

Sec 11.6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS REARRANGEMENTS Divergent convergent See page 719

Sec 11.6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS REARRANGEMENTS REMARK: Absolutely convergent any rearrangement has the same sum s with sum s Riemann proved that Conditionally convergent there is a rearrangement that has a sum equal to r. r is any real number

Sec 11.6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS TERM-091

Sec 11.6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS TERM-082

Sec 11.6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS TERM-082

Sec 11.6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS TERM-091