Direct comparison Test and Limit Comparison Test Direct

Direct comparison Test and Limit Comparison Test Direct Comparison Test If an and bn are the nth terms of two infinite series, and if we know the behavior of one of them, then the behavior of the other can be obtained using the direct comparison test, provided the following conditions are satisfied. If 0 < an bn for all values of n, then

Use direct comparison test to determine the convergence or divergence of the series an is the given series. We have to identify a series bn whose behavior we know and such that an < bn Since the degree of the denominator of an is 2, bn also must be of degree 2. The obvious choice is

Use direct comparison test to determine the convergence or divergence of the series What is a suitable bn to compare an such that an < bn The obvious choice is

Use direct comparison test to determine the convergence or divergence of the series What is a suitable bn to compare an? The obvious choice is Since p < 1, this is a diverging p-series. bn < an and bn diverges

Use direct comparison test to determine the convergence or divergence of the series What is a suitable bn to compare an such that an < bn The obvious choice is

Limit Comparison Test If an and bn are the nth terms of two infinite series, and if we know the behavior of one of them, then the behavior of the other can be obtained using the limit comparison test. If an > 0 and bn > 0 for all values of n and where L is finite and positive: Review: As n increases, the denominator grows without bounds. The numerator and the denominator have the same degree The limit is the ratio of the leading coefficient.

Use the limit comparison test to determine the convergence or divergence of the series What is a suitable bn whose behavior we know. The obvious choice is is a diverging harmonic series. This limit is finite and positive.

Use the limit comparison test to determine the convergence or divergence of the series What is a suitable bn whose behavior we know. The obvious choice is This limit is finite and positive is a diverging harmonic series.

Use the limit comparison test to determine the convergence or divergence of the series What is a suitable bn whose behavior we know. The obvious choice is This is a converging geometric series. This limit is finite and positive

Use the limit comparison test to determine the convergence or divergence of the series What is a suitable bn whose behavior we know. The obvious choice is This is a diverging harmonic series. This limit is finite and positive

Review of Tests so far 1. nth Term Test 2. Geometric Series Test 3. Telescopic Series Test 4. Integral Test 5. Direct Comparison Test 6. Limit Comparison Test

Test the convergence or divergence of the series: of the series Which of the tests is appropriate in this case? This is a geometric series with a = 5 and This is a converging geometric series. Can you find the sum of this converging geometric series? Answer:

Use the limit comparison test to determine the convergence or divergence of the series Use limit comparison test What is a suitable bn whose behavior we know. The obvious choice is This limit is finite and positive. is a converging p-series.

Test the convergence or divergence of the series: What is an appropriate test in this case? When you are in doubt, always try the nth term test? Since the limiting value of the nth term is not zero, this series diverges.

Test the convergence or divergence of the series: of the series What is an appropriate test in this case? Since the corresponding function can be integrated, we should try integral test. The corresponding function is: x 2 + 1 = u 2 xdx = du/2 Since the integral converges, the series converges.

Test the convergence or divergence of the series: of the series What is an appropriate test in this case? Use partial fractions to simplify Equate the numerator and denominator. 3 = A(n + 3) + Bn Setting n = -3 gives: This is a converging telescopic series. Setting n = 0 gives: A=1 B = -1

Given that: