SECTION 8.3 THE INTEGRAL AND COMPARISON TESTS

P2P28.3 INFINITE SEQUENCES AND SERIES  In general, it is difficult to find the exact sum of a series. We were able to accomplish this for geometric series and the series ∑1/[n(n+1)]. This is because, in each of these cases, we can find a simple formula for the nth partial sum s n. Nevertheless, usually, it isn ’ t easy to compute

P3P38.3 INFINITE SEQUENCES AND SERIES  Therefore, in this section and the next we develop tests that enable us to determine whether a series is convergent or divergent without explicitly finding its sum.  In this section we deal only with series with positive terms, so the partial sums are increasing.  In view of the Monotonic Sequence Theorem, to decide whether a series is convergent or divergent, we need to determine whether the partial sums are bounded or not.

P4P48.3 TESTING WITH AN INTEGRAL  Let ’ s investigate the series whose terms are the reciprocals of the squares of the positive integers: There ’ s no simple formula for the sum s n of the first n terms.

P5P58.3 INTEGRAL TEST  However, the computer-generated values given here suggest that the partial sums are approaching near 1.64 as n → ∞. So, it looks as if the series is convergent. We can confirm this impression with a geometric argument.

P6P68.3 INTEGRAL TEST  We can confirm this impression with a geometric argument.  Figure 1 shows the curve y = 1/x 2 and rectangles that lie below the curve.

P7P78.3 INTEGRAL TEST  The base of each rectangle is an interval of length 1.  The height is equal to the value of the function y = 1/x 2 at the right endpoint of the interval.

P8P88.3 INTEGRAL TEST  Thus, the sum of the areas of the rectangles is:

P9P98.3 INTEGRAL TEST  If we exclude the first rectangle, the total area of the remaining rectangles is smaller than the area under the curve y = 1/x 2 for x ≥ 1, which is the value of the integral In Section 6.6, we discovered that this improper integral is convergent and has value 1.

P108.3 INTEGRAL TEST  So, the image shows that all the partial sums are less than Therefore, the partial sums are bounded.  We also know that the partial sums are increasing (as all the terms are positive).  Thus, the partial sums converge (by the Monotonic Sequence Theorem). So, the series is convergent.

P118.3 INTEGRAL TEST  The sum of the series (the limit of the partial sums) is also less than 2:

P128.3 INTEGRAL TEST  The exact sum of this series was found by the mathematician Leonhard Euler (1707 – 1783) to be  2 /6. However, the proof of this fact is beyond the scope of this book.

P138.3 INTEGRAL TEST  Now, let ’ s look at this series:

P148.3 INTEGRAL TEST  The table of values of s n suggests that the partial sums aren ’ t approaching a finite number. So, we suspect that the given series may be divergent. Again, we use a picture for confirmation.

P158.3 INTEGRAL TEST  Figure 2 shows the curve y = 1/.  However, this time, we use rectangles whose tops lie above the curve.

P168.3 INTEGRAL TEST  The base of each rectangle is an interval of length 1.  The height is equal to the value of the function y = 1/ at the left endpoint of the interval.

P178.3 INTEGRAL TEST  So, the sum of the areas of all the rectangles is:

P188.3 INTEGRAL TEST  This total area is greater than the area under the curve y = 1/ for x ≥ 1, which is equal to the integral However, we know from Section 6.6 that this improper integral is divergent. In other words, the area under the curve is infinite.

P198.3 INTEGRAL TEST  Thus, the sum of the series must be infinite. That is, the series is divergent.  The same sort of geometric reasoning that we used for these two series can be used to prove the following test. The proof is given at the end of the section.

P208.3 Suppose f is a continuous, positive, decreasing function on [1, ∞ ) and let a n = f(n). Then, the series is convergent if and only if the improper integral is convergent. In other words, (a) If is convergent, then is convergent. (b) If is divergent, then is divergent. THE INTEGRAL TEST

P218.3 NOTE  When we use the Integral Test, it is not necessary to start the series or the integral at n = 1. For instance, in testing the series we use

P228.3 NOTE  Also, it is not necessary that f be always decreasing. What is important is that f be ultimately decreasing, that is, decreasing for x larger than some number N. Then, is convergent. So, is convergent by Note 4 of Section 8.2