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Summing inverses II (Posted on 2012-03-16) Difficulty: 3 of 5
The sum of the reciprocal of the square root of all the positive integers up to n is denoted by F(n), that is:

F(n) = 1+1/√2 + 1/√3 +...+ 1/√n

Determine the maximum value of n such that the integer part of the base ten expansion of F(n) DOES NOT exceed 2012.

*** For an extra challenge, solve this puzzle without using a computer program.

No Solution Yet Submitted by K Sengupta    
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Hints/Tips Hint Comment 12 of 12 |
(In reply to solution by Dej Mar)

"Euler's Summation Formula" as given  in "Numerical Mathematical Analysis" by  James B Scarborough, Sixth Edition, The Johns Hopkins Press (page 166) makes for a simplified procedure for deriving sums of similar expressions having a large number of terms.

Took a while to search the net to observe that the same formula is given in Wikipedia under the article "Euler–Maclaurin formula" and, in Mathworld under the article "Euler-Maclaurin Sum Formula" (which redirects to "Euler-Maclaurin Integration Formulas")

Members may use the above formula to achieve a greater degree of accuracy in determining the desired value of n.

Edited on March 19, 2012, 2:50 am
  Posted by K Sengupta on 2012-03-19 02:39:43

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