Section 11.5 – Testing for Convergence at Endpoints

There are lots of other tests we can use to test for convergence. The Integral Test This is useful in making decisions on series such as:

Another nice test: The Alternating Series Test Harmonic is a special case of this. Another nice test: The Alternating Series Test We can use all of these facts to test the endpoints of our interval of convergence.

Find the interval of convergence for converges, alternating series test If x = -1, diverges, harmonic series If x = 1, Interval of convergence [-1, 1)

Find the interval of convergence for If x = -7, diverges, nth term test If x = 1, diverges, nth term test Interval of convergence (-7, 1)

Find the interval of convergence for

Find the interval of convergence for If x = 2, diverges, harmonic series If x = 4, converges, alternating series test Interval of convergence (2, 4]

We can also compare a new series to one we already know. Harmonic Series - DIVERGES p-Series Converges if p > 1 Diverges if p < 1 Comparison Test for Convergence Comparison Test for Divergence

acts like for all n converges by the comparison test to the p-series with p = 2. acts like diverges by the comparison test to the p-series with p = 1. (Harmonic)

diverges by the comparison test to haromonic series. diverges by the nth term test for divergence.

We say… Converges Conditionally Converges Absolutely So…

Determine whether the series is converges conditionally, converges absolutely, or diverges.

Determine whether the series is converges conditionally, converges absolutely, or diverges. Alternates between -1 and 1. Diverges. Converges

Determine whether the series is converges conditionally, converges absolutely, or diverges. The series diverges by the nth term test for divergence We could have looked at the absolute value and then the function itself, but we would get the same result.

Determine whether the series is converges conditionally, converges absolutely, or diverges.

Determine whether the series is converges conditionally, converges absolutely, or diverges. Both alternate… Since infinite geometric with r = 3/5, the series converges absolutely.

OR Determine whether the series is converges conditionally, converges absolutely, or diverges. OR

We can approximate the value of the sum of an alternating series if… Error Estimating We can approximate the value of the sum of an alternating series if… If we approximate the sum of the first n terms, we will be off by no more than the absolute value of the next term.

Find an upper bound for the error if the sum of the first four terms is used as an approximation to the sum of the series.

Find the smallest value of n for which the nth partial sum approximates the sum of the series within 0.005.

Holy Cow… Lots to remember. Here’s a chart that might help. Test Series Converge/Diverge Comments nth-term Geometric Integral The function f obtained must be continuous, positive, and decreasing. p-series Alternating Applicable only to an alternating series Useful for series that contain both positive and negative terms.