Math 2414 Section 8 6 Alternating Series Our previous discussion focused on infinite series with positive terms There are many interesting series with terms of mixed sign For example the series 1 1 1 1 1 1 1 1 K 2 3 4 5 6 7 8 has the pattern that two positive terms are followed by two negative terms and vice versa Infinite series can have a variety of sign patterns so we need to restrict our attention The simplest sign pattern is also the most important We consider alternating series in which the signs strictly alternate as in the series 1 k 1 The factor 1 k 1 or 1 k k 1 1 1 1 1 1 1 1 1 K 2 3 4 5 6 7 8 k has the pattern K 1 1 1 1 K and provides the alternating signs Alternating Harmonic Series k 1 1 Consider the series k 1 k which is called the alternating harmonic series Recall that this 1 series without the alternating signs k 1 k is the divergent harmonic series Does adding alternating signs change the convergence or divergence of a series Alternating Series 1 Alternating series in general are written 1 alternating signs are provided by k 1 or k ak 1 or k 1 ak where ak 0 The k 1 With the exception of the Divergence Test none of the convergence tests for series with positive terms applies to alternating series The fortunate news is that only one test needs to be used for alternating series and it is easy to use Theorem 8 18 The Alternating Series Test k 1 k 1 ak 1 ak The alternating series or converges provided a s 1 the k are positive 2 the ak s are nonincreasing lim ak 0 3 k 0 ak 1 ak for k greater than some index N and There is potential for confusion For series of positive terms lim ak 0 does not imply lim ak 0 convergence For alternating series with nonincreasing terms k does imply convergence k Theorem 8 19 Alternating Harmonic Series k 1 1 1 1 1 1 1 K 2 3 4 5 The alternating harmonic series k 1 k converges even though the 1 1 1 1 1 1 K 2 3 4 5 harmonic series k 1 k diverges Determine whether the following series converge or diverge n 1 n n 1 Determine whether the following series converge or diverge 3 4 5 2 K 2 3 4 1 k 2 k k ln k Determine whether the following series converge or diverge cos np n 2 n Absolute and Conditional Convergence Removing the alternating signs in a convergent series may or may not result in a convergent series The terminology that we now introduce distinguishes these cases Definition Absolute Convergence The infinite series a k k 1 converges absolutely i e is absolutely convergent if the corresponding series of absolute values a k k 1 converges Definition Conditional Convergence A series that converges but does not converge absolutely converges conditionally i e is conditionally convergent 1 The series k 1 k2 is an example of an absolutely convergent series because the series of absolute values 1 k 1 k2 k 1 1 2 k 1 k is a convergent p series In this case removing the alternating signs in the series does not affect its convergence k 1 1 k On the other hand the convergent alternating harmonic series has the property that the corresponding series of absolute values 1 k 1 k k 1 1 k 1 k does not converge In this case removing the alternating signs in the series does affect convergence so this series does not converge absolutely Instead we say it converges conditionally A convergent series may not converge absolutely It is however true that if a series converges absolute then it converges Theorem 8 21 Absolute Convergence Implies Convergence a a If k converges then k converges absolute convergence implies convergence If ak diverges then ak diverges Determine whether the following series diverge converge absolutely or converge conditionally 1 3 n 1 n 1 k 1 k 1 k Determine whether the following series diverge converge absolutely or converge conditionally 1 k 1 k 1 k3 sin k k k 1 2 Determine whether the following series diverge converge absolutely or converge conditionally k 1 k k 1 k 1 Guidelines Here is a reasonable course of action when testing a series for convergence 1 Divergence Test 2 Apply absolute values and move to the next step Remember if the series converges it does so absolutely If the series does not converge you must test for conditional convergence 3 Is the resulting series a series for which we can determine the sum a Geometric series b Telescoping series 4 Is the resulting series a p series 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 the 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 3 and 4 as comparison series 8 If the kth term can by integrated try the Integral Test 9 If any of the tests in Steps 3 7 show the series to be divergent try the Alternating Series Test If the series converges it does so conditionally Table 8 4 Special Series and Convergence Tests Condition for Condition for Series or test Form of series Convergence Divergence Comments Cannot be used k lim a 0 a k Divergence Test Does not apply to prove k k 1 convergence r 1 If then k ar r 1 r 1 Geometric Series a ar k k 0 1 r k 0 a k where ak f k and f is continuous positive and decreasing 1 p k 1 k k 1 Integral Test p series f x dx 1 f x dx 1 does not exist p 1 p 1 Useful for comparison tests lim ak 1 1 k a k lim ak 1 1 k a k Inconclusive if a lim k 1 1 k a k lim k ak 1 lim k ak 1 Inconclusive if lim k ak 1 Ratio Test a k k 1 where ak 0 Root Test a k where ak 0 k 1 The value of the integral is not the value of the series k k k Direct Comparison Test a k where ak 0 k 1 0 ak bk and b and k k 1 0 bk ak converges b k k 1 diverges a k is given you supply k 1 b k k 1 Limit Comparison Test a k where ak 0 bk 0 k 1 ak k b k 0 lim b k and k 1 converges a lim k 0 k b k and b k k 1 a k is given you supply k 1 diverges b k k 1 …