**9.5 Testing Convergence at Endpoints** Petrified Forest National Park, Arizona

**If is a series with positive terms and** then: The series converges if . The series diverges if . The test is inconclusive if . Remember: The Ratio Test:

**This section in the book presents several other tests or** techniques to test for convergence, and discusses some specific convergent and divergent series.

**If is a series with positive terms and** then: The series converges if . The series diverges if . The test is inconclusive if . Nth Root Test: Note that the rules are the same as for the Ratio Test.

**example:** ?

**formula #104** formula #103 Indeterminate, so we use L’Hôpital’s Rule

**it converges** example: ?

**it diverges** another example:

**The Integral Test** If is a positive sequence and where is a continuous, positive decreasing function, then: and both converge or both diverge. Remember that when we first studied integrals, we used a summation of rectangles to approximate the area under a curve: This leads to:

**Example 1:** Does converge? Since the integral converges, the series must converge. (but not necessarily to 2.)

**converges if , diverges if .** p-series Test We could show this with the integral test. If this test seems backward after the ratio and nth root tests, remember that larger values of p would make the denominators increase faster and the terms decrease faster.

**It diverges very slowly, but it diverges.** Because the p-series is so easy to evaluate, we use it to compare to other series. the harmonic series: diverges. (It is a p-series with p=1.)

**If , then both ** and converge or both diverge. If , then converges if converges. If , then diverges if diverges. Limit Comparison Test If and for all (N a positive integer)

**Since diverges, the series diverges.** Example 3a: When n is large, the function behaves like: harmonic series

**Since converges, the series converges.** Example 3b: When n is large, the function behaves like: geometric series

**Alternating Series Test** If the absolute values of the terms approach zero, then an alternating series will always converge! Alternating Series The signs of the terms alternate. Good news! example: This series converges (by the Alternating Series Test.) This series is convergent, but not absolutely convergent. Therefore we say that it is conditionally convergent.

**Alternating Series Estimation Theorem** For a convergent alternating series, the truncation error is less than the first missing term, and is the same sign as that term. Since each term of a convergent alternating series moves the partial sum a little closer to the limit: This is a good tool to remember, because it is easier than the LaGrange Error Bound.

**F3** 4 There is a flow chart on page 505 that might be helpful for deciding in what order to do which test. Mostly this just takes practice. To do summations on the TI-89: becomes becomes

**ENTER** ENTER GRAPH Y= WINDOW To graph the partial sums, we can use sequence mode. MODE Graph……. 4

**ENTER** ENTER GRAPH Y= WINDOW Table To graph the partial sums, we can use sequence mode. MODE Graph……. 4

**ENTER** ENTER GRAPH Y= WINDOW Table To graph the partial sums, we can use sequence mode. MODE Graph……. 4 p