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Is $\lim\limits_{k\to\infty}\sum\limits_{n=k+1}^{2k}{\frac{1}{n}} = 0$?
I need to Prove the following sum converges:
$$\lim_{n\to\infty}\sum\limits_{i=1}^n\dfrac{1}{n+i}$$
What methods can I use?
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CodeKingPlusPlus
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Comparison of sums and integrals. – Did Sep 29 '12 at 20:59
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Are you taking the limit for $n\to \infty$? – Jean-Sébastien Sep 29 '12 at 21:00
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Yes, I will specify that – CodeKingPlusPlus Sep 29 '12 at 21:01
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2Language note: This is not a "sum convergence" problem. We say a sum converges of the sequence $a_0$, $a_0+a_1$, $a_0+a_1+a_2$,... converges. This problem is not a problem of that type. Rather this is a question about the limit of a sequence of different (related) sums. – Thomas Andrews Sep 29 '12 at 21:24
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Hint: you can rewrite your sum as $$ \lim_{n\to\infty} \frac{1-0}{n}\sum_{i=1}^n \frac{1}{1+\frac{i}{n}}, $$ Now do you know the definition of Riemann's integral?
Added: Somehow the english Wiki page doesn't not seem to show this as explicitly as the french one does, but you can have a look at this page, the first section shows what you have, with $f$ replaced by $\frac{1}{1+x}$
Jean-Sébastien
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Is Riemann's integral that theorem they teach you in calc 1 when you take the number of rectangles under a curve to be infinite and compute their areas? If so, it has been a long long time! hahah – CodeKingPlusPlus Sep 29 '12 at 21:09
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It is somewhat the way Riemann's integral is constructed in the first calculus course, via approximatio of areas under curves by rectangle. – Jean-Sébastien Sep 29 '12 at 21:12
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HINT: The sequence of sums is bounded above by $1$: $$\sum_{i=1}^n\dfrac{1}{n+i}<\sum_{i=1}^n\frac1n=1\;.$$
It’s also strictly increasing, as you can show by calculating
$$\sum_{i=1}^{n+1}\dfrac{1}{n+1+i}-\sum_{i=1}^n\dfrac{1}{n+i}=\frac1{2(n+1)}+\sum_{i=1}^n\left(\frac1{n+1+i}-\frac1{n+i}\right)\;;$$
I’ll leave the rest of that calculation to you. Note that the last sum telescopes.
Brian M. Scott
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What does this last equation show? You are finding the $(n+1)th$ term correct? – CodeKingPlusPlus Sep 29 '12 at 21:59
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@CodeKingPlusPlus: No, if $s_n$ is the $n$-th term, I’m calculating $s_{n+1}-s_n$. What kind of result would show that the sequence of $s_n$’s is increasing? – Brian M. Scott Sep 29 '12 at 22:01
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Ok, so if the difference of $s_{n+1} - s_n > 0$ then $s_n$ is increasing. – CodeKingPlusPlus Sep 29 '12 at 22:07
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Hint
$\frac{1}{n+i}=\frac{1}{n\left(1+\frac{i}{n}\right)}$, then rewrite sum $\sum\limits_{i=1}^n\dfrac{1}{n+i}$ as Riemann sum for appropriate integral.
M. Strochyk
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$$\sum_{i=1}^n\frac{1}{n+i}=\frac{1}{n}\sum_{i=1}^n\frac{1}{1+\frac{i}{n}}\xrightarrow [n\to\infty]{} \int_0^1\frac{dx}{1+x}=...$$
DonAntonio
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Could you show some derivation of how you get $\int\dfrac{dx}{1+x}$ – CodeKingPlusPlus Sep 29 '12 at 21:34
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Riemann sums, as simple as that. Since the function $,f(x)=\frac{1}{1+x},$ is continuous in $,[0,1],$ its Riemann integral exists there and we can thus choose any subdivision of $,[0,1],$ we want to form the Riemann sums. We choose $,{x_0=0,,,x_1=1/n,,,x_2=2/n,,...,,x_n=n/n=1},$ and we form the Riemann sum $$\sum_{i=1}^nf(x_i)(x_i-x_{i-1})=\sum_{i=1}^n\frac{1}{1+\frac{i}{n}}\cdot\frac{1}{n}$$ – DonAntonio Sep 29 '12 at 21:41