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   Author  Topic: Summation of series  (Read 2758 times)
Optimus Prime
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Summation of series  
« on: Jan 21st, 2008, 8:19pm »
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Find the sum of the first n terms of the following series:
1/7 + 1/77 + 1/777 + ... up to n-th term  
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ThudnBlunder
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Re: Summation of series  
« Reply #1 on: Jan 21st, 2008, 10:23pm »
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So we need to find  
        n  
(9/7)1/(10k - 1)  
       k=1
« Last Edit: Jan 22nd, 2008, 7:56am by ThudnBlunder » IP Logged

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Icarus
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Re: Summation of series  
« Reply #2 on: Jan 24th, 2008, 5:13pm »
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If Optimus Prime has a nice answer for this, it would be interesting from a number theory point-of-view.
 
If we let x = 0.1, we can rewrite T&B's summation as  
k=1..n xk/(1 - xk) = k=1..n j=1.. xkj = m=1.. d(m; n) xm,
 
where d(m; n) = |{ d : d | m  and  d <= n}| is the number of divisors of m less than or equal to n.
 
Thus a generalized solution to this problem could provided a means of generating the values of d(m; n).
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