Sec 11 6 ABSOLUTE CONVERGENCE AND THE RATIO

Sec 11. 6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS DEF: REMARK: Is called Absolutely convergent IF convergent Example: Test the series for absolute convergence. Notice that if a series with positive terms, then absolute convergence is the same as convergence. Example: Test the series for absolute convergence.

Sec 11. 6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS Example: DEF: Is called Absolutely convergent Test the series for absolute convergence. IF convergent Example: DEF: Is called conditionally convergent if it is convergent but not absolutely convergent. Test the series for absolute convergence.

Sec 11. 6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS THM: Absolutely convergent Example: Determine whether the series converges or diverges. convergent

Sec 11. 6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS THM: Absolutely convergent

Sec 11. 6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS Series Tests 1) 2) 3) 4) 5) 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: 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 1) 2) 3) 4) 5) 6) Example: Test for Divergence Integral Test Comparison Test Limit Comparison Test Alternating Test Ratio Test Root Test Let Test the series for convergence.

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 r is any real number there is a rearrangement that has a sum equal to r.

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