3 TESTS Sec 11.3: THE INTEGRAL TEST Sec 11.4: THE COMPARISON TESTS Sec 11.4: THE LIMIT COMPARISON TESTS

Sec 11.3: THE INTEGRAL TEST

Sec 11.3: THE INTEGRAL TEST Example: Test the series for convergence or divergence.

THE INTEGRAL TEST AND ESTIMATES OF SUMS THEOREM: (Integral Test) Remark: a continuous, positive, decreasing function on [1, inf) Convergent Convergent sequence of positive terms. Divergent Dinvergent Example: Solution: Test the series for convergence or divergence. Since this improper integral is convergent, the series is also convergent by the integral test.

THE INTEGRAL TEST AND ESTIMATES OF SUMS THEOREM: (Integral Test) a continuous, positive, decreasing function on [1, inf) Convergent Convergent Divergent Dinvergent Example: Solution: Test the series for convergence or divergence. Since this improper integral is divergent, the series is also divergent by the integral test.

THE INTEGRAL TEST AND ESTIMATES OF SUMS THEOREM: (Integral Test) a continuous, positive, decreasing function on [1, inf) Convergent Convergent Divergent Dinvergent Special Series: Example: Geometric Series Harmonic Series Telescoping Series p-series Alternatingp-series Harmonic Series is the series convergent?

THE INTEGRAL TEST AND ESTIMATES OF SUMS TERM-102

Memorize: THE INTEGRAL TEST AND ESTIMATES OF SUMS Example: Special Series: Geometric Series Harmonic Series Telescoping Series p-series Alternatingp-series Example: For what values of p is the series convergent?

THE INTEGRAL TEST AND ESTIMATES OF SUMS Example: P Series: For what values of p is the series convergent? Example: Example: Test the series for convergence or divergence. Test the series for convergence or divergence.

THE INTEGRAL TEST AND ESTIMATES OF SUMS FINAL-081

THE INTEGRAL TEST AND ESTIMATES OF SUMS THEOREM: (Integral Test) a continuous, positive, decreasing function on [1, inf) Convergent Convergent Divergent Dinvergent REMARK: REMARK: We should not infer from the Integral Test that the sum of the series is equal to the value of the integral. In fact, Integral Test just test if convergent or divergent. But if it is convergent what is the sum??

Sec 11.4: THE COMPARISON TESTS

Sec 11.4: THE COMPARISON TESTS In the comparison tests the idea is to compare a given series with a series that is known to be convergent or divergent. THEOREM: (THE COMPARISON TEST) convg Known Series Example: Determine whether the series converges or diverges. geometric Solution: P-series

Sec 11.4: THE COMPARISON TESTS In the comparison tests the idea is to compare a given series with a series that is known to be convergent or divergent. THEOREM: (THE COMPARISON TEST) convg Known Series Example: Determine whether the series converges or diverges. geometric Solution: P-series

Sec 11.4: THE COMPARISON TESTS In the comparison tests the idea is to compare a given series with a series that is known to be convergent or divergent. THEOREM: (THE COMPARISON TEST) convg THEOREM: (THE COMPARISON TEST) divg

Sec 11.4: THE COMPARISON TESTS In the comparison tests the idea is to compare a given series with a series that is known to be convergent or divergent. THEOREM: (THE COMPARISON TEST) divg Example: Determine whether the series converges or diverges. Solution:

THE LIMIT COMPARISON TESTS Sec 11.4: THE LIMIT COMPARISON TESTS

Sec 11.4: THE COMPARISON TESTS THEOREM: (THE LIMIT COMPARISON TEST) either both series converge or both diverge. With positive terms Example: Example: Solution: Solution: Determine whether the series converges or diverges. Determine whether the series converges or diverges. REMARK: Notice that in testing many series we find a suitable comparison series by keeping only the highest powers in the numerator and denominator.