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I have been trying to determine a general formula for the following summation for awhile now and cannot seem to make any progress:

$$\sum_{k=1}^{n} 1/(n+k)$$

In a sense, what I am asking is similar to determining a summation for, say, the sum of all integers $1 + 2 + 3 + ... + n$, which can be rewritten as

$$\sum_{i=1}^{n} i$$

Which can be shown is equal to $$ n(n+1)/2$$

Therefore my question really becomes, how might I approach a question where there are these two variables now, n and k, in order to derive a general formula?

user
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HtmlProg
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  • Possibly relevant, if not outright duplicate: https://math.stackexchange.com/questions/52572/do-harmonic-numbers-have-a-closed-form-expression – SK19 Sep 28 '19 at 23:04
  • That is $H_{2n}-H_n$, which is convergent to $\log(2)=\int_{1}^{2}\frac{dx}{x}$ by Riemann sums or alternative arguments. – Jack D'Aurizio Sep 29 '19 at 13:46

1 Answers1

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HINT

Note that

$$\sum_{k=1}^{n} 1/(n+k)=\sum_{k=1}^{2n} 1/k-\sum_{k=1}^{n} 1/k$$

user
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  • I was under the impression that $H_n=\sum_{k=1}^n\frac{1}{k}$ has no closed form? – SK19 Sep 28 '19 at 22:57
  • @SK19 Yes of course, the result is $H_{2n}-H_n$ that is $\approx \ln 2$ for $n$ large. – user Sep 28 '19 at 23:00
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    @SK19 My hint is indeed just a hint to start with the solution, aimed for the main doubt: "I approach a question where there are these two variables". Morover, I'm not sure the OP is aware about harmonic numbers. – user Sep 28 '19 at 23:11