9Remember that when we first studied integrals, we used a summation of rectangles to approximate the area under a curve:This leads to:The Integral TestIf is a positive sequence and whereis a continuous, positive decreasing function, then:and both converge or both diverge.
10Example 1:Does converge?Since the integral converges, the series must converge.(but not necessarily to 2.)
11p-series Test converges if , diverges if . 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.
12the harmonic series:diverges.(It is a p-series with p=1.)It diverges very slowly, but it diverges.Because the p-series is so easy to evaluate, we use it to compare to other series.
13Limit Comparison Test If and for all (N a positive integer) If , then both andconverge or both diverge.If , then converges if converges.If , then diverges if diverges.
14Example 3a:When n is large, the function behaves like:harmonic seriesSince diverges, the series diverges.
15Example 3b:When n is large, the function behaves like:geometric seriesSince converges, the series converges.
16Good news! Alternating Series Test Alternating Series The signs of the terms alternate.If the absolute values of the terms approach zero, then an alternating series will always converge!Alternating Series TestGood 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.
17Alternating Series Estimation Theorem Since each term of a convergent alternating series moves the partial sum a little closer to the limit:Alternating Series Estimation TheoremFor a convergent alternating series, the truncation error is less than the first missing term, and is the same sign as that term.This is a good tool to remember, because it is easier than the LaGrange Error Bound.
18There 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:becomesF34becomes
19To graph the partial sums, we can use sequence mode. 4ENTERY=ENTERWINDOWGRAPH
20To graph the partial sums, we can use sequence mode. 4ENTERY=ENTERWINDOWGRAPHTable
21p To graph the partial sums, we can use sequence mode. Graph……. 4 Y= ENTERY=ENTERWINDOWGRAPHTablep