Convergence of an Alternating SeriesDate: 02/23/98 at 21:57:19 From: Harris Welt Subject: convergence of an alternating series Dear Dr. Math, I'm trying to figure out whether or not this series is convergent. Computing the sum from n = 1 to infinity, (-1)^(n+1)*(ln (n)/n), is the series convergent if the limit of (ln (n)/n) is convergent? If so, how can this limit be found? If not, why not? Also, if you get a chance, explain why this works. I've been having a lot of trouble understanding my math teacher in this area. Thanks, Harris Date: 02/24/98 at 09:34:25 From: Doctor Anthony Subject: Re: convergence of an alternating series For an alternating series, you will ALWAYS have convergence if the limit of u(r) as r -> infinity is zero. So this series converges if: Lt ln(n)/n = 0 n->infin. Using l'Hopital's rule as n->infinity ln(n) 1/n ------ = ------ = 1/n -> 0 as n -> infinity. n 1 You can see why the alternating series converges if you plot S1, S2, S3, etc on a number line. S1 is positive and goes to the right. S2 is the result of subtracting a term from S1 so brings you to the left a bit, S3 is the result of adding a term to S2 so takes you to the right again, but not as far as S1. The points will jump to and fro about the final position, S, which is less than S1 but greater than S2. | 0-------------S2------S4---S6--|--S7---S5----S3---------S1 | S -Doctor Anthony, The Math Forum http://mathforum.org/dr.math/ |
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