Study Guide-1.pdf - Math 21C Study Guide For Midterm 1...

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Math 21C Study Guide For Midterm 1April 17, 2018Here’s a small study guide on the convergence tests you’ve gone over in the past week.First, let’s look at some inequalities that will help us when calculating limits. Leta(n) andb(n) be two functions that depend onn. When I writea(n)<< b(n), I mean limn→∞a(n)b(n)=0, i.e.b(n) grows a lot more quickly thana(n). With this, we get a nice comparison betweencommon expressions we’ll see in this course:nn>> n!>> xn>> ny>>lnn,wherexis any real number bigger than 1, andyis any real number bigger than 0. So, forexample,limn→∞(1.01)nn100=.We can see this by using L’Hospital’s rule a bunch of times. The one-hundredth derivative of(1.01)nis (1.01)n(ln(1.01))100, while the one-hundredth derivative ofn100is 100!. That’s abig number but it doesn’t grow, and limn→∞(1.01)n=,and so by L’Hospital’s rule,limn→∞(1.01)nn100= limn→∞(1.01)n(ln(1.01))100100!=.nth-term test (aka the test for divergence)What it says:Thenth term test states that given a seriesn=1an, if limn→∞an6= 0,then the series diverges. Otherwise, the test is inconclusive.Otherwise, the test is inconclusive..Otherwise, the test is inconclusive...1
I repeated the last sentence because it gets ignored pretty often, even though it is superimportant. This test willnevertell you that a series converges. It will tell you either thatthe series diverges, or it will tell you that it has no idea what’s going on.When to use it:It’s never a bad idea to use this test first when encountering a series,since it can end saving you some work/time. If you find yourself super confused as to howto show whether or not some weird series converges, you should probably use this test. It’slikely that the terms of the series don’t go to 0. There are some really good examples ofthis in the homework, but I won’t say which ones those are, at least not until section.Integral TestWhat is says:Suppose we’re given a seriesn=Nan, and we can writean=f(n), wherefis a continuous, positive, anddecreasingfunction on (N,).Then,n=NanandRNf(x)dxboth converge or both diverge.Basically, if you have a seriesn=Nanand the terms look like some functionf(x) thatyou can integrate, then you can check if the series converges by checking if the integralRNf(x)dxconverges.When to use it:You generally want to use this test when the terms of the series look likea continuous function you can integrate when you replace then’s withx’s. Oftentimes (butnot always!), you’ll want to use this test when the series has logarithms or exponentials inits terms. Some examples areXn=1nen2,Xn=21nlnn,Xn=21n(lnn)2.Note, these series correspond to the integrals in 1(c)-(e) of the previous worksheet. You’llwant to knowu-substitution and integration by parts, so review those topics if you needto. Because you need to remember a lot of old material, I generally try to see if I can usea different test for convergence. Sometimes, the integral test will require you to calculate

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