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Q: Approximate or Compute a finite harmonic series ( No Answer,   2 Comments )
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 Subject: Approximate or Compute a finite harmonic series Category: Science > Math Asked by: tc2city-ga List Price: \$5.00 Posted: 05 Nov 2006 06:17 PST Expires: 05 Dec 2006 06:17 PST Question ID: 780243
 ```Is their a formula to compute a finte harmonic series? 1+1/2+1/3+1/4...+1/n For example, if n=75, we can calculate the sums to be 4.9014. Could be used to solve this teaser: You are going to drive 75 miles. You are going to drive the 1st mile at 1 mph, The 2nd mile at 2 mph, The 3rd mile at 3 mph, on and on, until finally, the 75th mile at 75mph. How long will it take you to go the 75 miles?``` Clarification of Question by tc2city-ga on 05 Nov 2006 06:24 PST ```Obviously, I know I can brute force compute the value once n is given. I'm looking for some type of formula where n is a variable.(ex. arithmetic series = n(n+1)/2)```
 ```You can get a good approximation by calculating ln n + .5772. The larger n is, the better the approximation. (ln n is the natural, or base-e, logarithm of n; .5772 is the approximate value of gamma, or Euler's Constant, which is the limit, as n increases without bound, of the sum of the series 1+1/2+1/3+...+1/n-ln n.) For your example with n = 75, the calculation evaluates to about 4.895, which is reasonably close to your value of 4.901.```
 ```The formula below gives an even better approximation: sum = ln n + gamma + 1/(2n) - 1/(12n^2) + 1/(120n^4) ... The formula above was found at: http://en.wikipedia.org/wiki/Euler-Mascheroni_constant Here is gamma to 45 decimal places, from the same webpage: ? ? 0.57721 56649 01532 86060 65120 90082 40243 10421 59335```