Sum of Harmonic SeriesDate: 5/9/96 at 23:17:50 From: Paulette M. Tarantola Subject: Sum of Harmonic Series My question pertains to what the total sum in terms of variables is for the series: (1/1*2)+(2/2*3)......+n/n(n+1) I am having difficulty finding the sum and do not know were to start. I would appreciate any help. Date: 11/13/96 at 08:04:38 From: Doctor Kuznicki Subject: Re: Sum of Harmonic Series Hi Paulette, First simplify your series equation: inf s = Sum 1/(n+1) n=1 [we know that this series diverges] Now write out the series (as you did above) but include the n-1 part: 1/2 + 1/3 + 1/4 + ... + 1/n + 1/(n+1) If you know the sum equation for one of these terms (1/n) then you can use this in your total sum: inf (n+1) Sum (1/n) = ln(n+1) [since 1 + 1/2 + 1/3 +...+1/n > Integral 1/x dx] n=1 1 Now use this in your equation: [Hint: replace (1/2 + 1/3 + 1/4 + ... + 1/n) with (ln(n+1) - 1)] s = ln(n+1) + 1/(n+1) - 1 The easiest way to solving these problems is by finding ways to use already known divergent series sum equations. One should also consider the other inequality: 1/2 + 1/3 + 1/4 + ... + 1/(n+1) < Integral (1 to n+1) [1/x.dx] = ln(n+1) So we have: ln(n+1) + 1/(n+1) -1 < 1/2 + 1/3 + .. + 1/(n+1) < ln(n+1) -Doctor Kuznicki, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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