Math 2414 Section 8 5 The Ratio Root and Comparison Tests We now consider several more convergence tests the Ratio Test the Root Test and two comparison tests The Ratio Test will be used frequently throughout the next chapter and comparison test are valuable when no other test works Again these tests determine whether an infinite series converges but they do not establish the value of the series The Ratio Test The Integral Test is powerful but limited because it requires evaluating integrals For example 1 k the series with a factorial term cannot be handled by the Integral Test The next test significantly enlarges the set of infinite series that we can analyze Theorem 8 14 The Ratio Test Let a ak 1 k a k r lim be an infinite series with positive terms and let 0 1 If r 1 the series converges 2 If r 1 including r the series diverges 3 If r 1 the test is inconclusive k So the Ratio Test says the limit of the ratio of the successive terms of the series must be less than 1 for convergence of the series Often you will see both the ratio and the root tests written with absolute values In such cases the requirement that the terms be positive becomes unnecessary In Section 8 6 we will further discuss the use of absolute values Lastly recall k k K 3 21 k 1 k 2 and 0 1 Use the Ratio Test to determine whether the following series converge 10k k 1 k Use the Ratio Test to determine whether the following series converge k k k 1 e kk k 1 k Use the Ratio Test to determine whether the following series converge 2k k k 1 3k k k 1 The Root Test Occasionally a series arises for which none of the preceding tests gives a conclusive result In these situations the Root Test may be the tool that is needed Theorem 8 15 The Root Test Let a be an infinite series with nonnegative terms and let 1 If 0 r 1 the series converges 2 If r 1 including r the series diverges 3 If r 1 the test is inconclusive k r lim k ak k Use the Root Test to determine whether the following series converge k 4k 2 3 2 7 k 6 k 1 Use the Root Test to determine whether the following series converge 2k 10 k 1 k k 1 1 2 k k k 1 Use the Root Test to determine whether the following series converge k k 1 2 k k 2 The Comparison Test Tests that use known series to test unknown series are called comparison tests The first test is the Direct Comparison Test or simply the Comparison Test Theorem 8 16 The Direct Comparison Test a b Let k and k be series with positive terms b a 1 If 0 ak bk and k converges then k converges b a 2 If 0 bk ak and k diverges then k diverges Whether a series converges depends on the behavior of terms in the tail large values of the index So the inequalities 0 ak bk and 0 bk ak need not hold for all terms of the series They must hold for all k N for some positive integer N The Direct Comparison Test is illustrated with graphs of sequences of partial sums Consider the series 1 a k 2 k 1 k 1 k 10 and 1 b k 2 k 1 k 1 k Consider the series k 4 k 4 ak 1 k 3 and Determine whether the following series converge k3 4 k 1 2 k 1 Determine whether the following series converge ln k 3 k 2 k k 4 k 4 bk 1 k n n 1 1 2 1 Determine whether the following series converge 1 2 k 1 k 3k 2 3k 2 2 2 k 1 k 3k 2 The Limit Comparison Test The Direct Comparison Test should be tried if there is an obvious comparison series and the k3 4 necessary inequality is easily established Notice however that if k 1 2k 1 had been k3 4 k 1 2k 10 then the comparison to the harmonic series would not work Rather than determining or manipulating inequalities it is often easier to use a more refined test called the Limit Comparison Test Theorem 8 17 The Limit Comparison Test ak L k b ak bk k Suppose that and are series with positive terms and a b 1 If 0 L that is L is a finite positive number then k and k either both converge or both diverge b a 2 If L 0 and k converges then k converges b a 3 If L and k diverges then k diverges lim Determine whether the following series converge k 4 2k 2 3 6 k 1 2k k 5 Determine whether the following series converge l 2 n 3 n 4 n n 1 l 2 1 Determine whether the following series converge k 2 k 2 k k 1 k 2 Guidelines a Here is reasonable course of action when testing a series of positive terms k for convergence 1 Divergence Test 2 Is the series a series for which we can determine the sum a Geometric series b Telescoping series 3 Is it a p series 4 Can the kth term be integrated Try the Integral Test k k 5 If the kth term involves k k or a where a is a constant try the Ratio Test 6 If k is only in an exponent try the Root Test 7 If the kth term is a rational function of k or a root of a rational function try the Direct Comparison Test or the Limit Comparison Test Use the families of series given in Steps 2 and 3 as comparison series
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