Divergence and Integral Test With the previous content, we used the idea of forming a closed form for the n th partial sum and taking its limit to determine the SUM of the series (if it exists). *** It is very rare to be able to construct a closed form, it is easy when you have (1) a geometric series or (2) a telescopic series Mostly, we will do some test to show the series will converge or diverge, and if it does converge we will then use a partial sum to find approximation for the series. The infinite series is written using sigma notation as: DIVERGENCE TEST: very useful to determine quickly whether an infinite series diverges. (a) If the converges, then lim u k better yet, we can say if the lim u k, then the (b) If the lim k u, then the test is inconclusive. (That is the series may converge or diverge) From (a), reality of these statements: If the infinite series converges, then each term of the series should be getting smaller and smaller, so it makes sense that the limit of the k th term goes to 0. (but in this statement, you must know that the series converges). Alternatively, if means the k th term of the series is not getting smaller, thus each term of the series will get bigger and bigger, thus the series diverges. Proof: Assume that the infinite series converges, then we can say that the sequence of partial sums must converge to a number, let's call this number, S. The infinite series can be viewed as: while the nth partial sum can be written as: re-arranging this, we can view the n th term of the series as the difference of two partial sums. *** For the infinite series, if lim u k Divergence Test Blank Page 1
Examples using the divergence test Example: Use the divergence test to determine whether the following series diverges or whether the test is inconclusive. (a) (b) (c) The Integral Test: Let be a series of positive terms ( where (decreasing term by term). Let, then the function is positive and decreasing on the closed interval. Then we can say that the infinite series,, and the improper integral,, ACT ALIKE. That is, if the integral diverges, then the series diverges. Likewise, if the integral converges, then the series converges. Note, this does NOT tell us what the series converges to, it only tells us that the series converges because the integral converges. Divergence Test Blank Page 2
Divergence Test Blank Page 3 Using the Integral Test Harmonic Series: this series looks harmless enough and it shows up a great deal in practice. We are going to show that this series diverges. You can see that the Divergence Test is inconclusive because so we need another approach to determine how this series behaves. Can we use the integral test? We need to be certain that the hypothesis of the integral test are valid: 1) are we dealing with a series of positive terms? 2) are the terms decreasing term by term? Create the improper integral and evaluate. Example: Determine whether the following series converges or diverges.
Divergence Test Blank Page 4 P-series The series given by: Converges if Diverges if Proof of this is based on the integral test (left to you). Estimation of the Sum of a convergent infinite series We can show is convergent, and now we want to find an approximation, S, of the series, We are going to use knowledge of the infinite series being made up of the n th partial sum PLUS many other sums of terms. That is, Our approach is to estimate the size of the remainder. That is, the remainder associated with the n th partial sum can be described as follows: R n =
Divergence Test Blank Page 5 Find an approximation and estimating the error Example: Consider the infinite series given by (a) Find an approximation of the 10 th partial sum,. Then estimate the error in using as an approximation to the sum of the series. (b) Now, find the value of n, such that is within 0.00001 of the sum of the series.
Divergence Test Blank Page 6 One more estimation example Example: Consider the infinite series given by (a) Find an approximation of the 25 th partial sum,. Then estimate the error in using as an approximation to the sum of the series. (b) Now, find the value of n, such that is within 0.00001 of the sum of the series.