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How do I evaluate $$\lim_{n \to +\infty}\frac{1}{n}\left({\frac{n}{n+1} + \frac{n}{n+2} + \cdots + \frac{n}{2n}}\right)\ ?$$

I'm stuck. I tried using the sandwich theorem but I was getting nowhere. Since $n \lt n+k \lt 2n$, I think it should be $ \infty$ .

rubik
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N S
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5 Answers5

14

It can be rewritten as

$$ \frac{1}{n} \sum_{k=1}^{n} \frac{n}{n+k} $$

$$ = \frac{1}{n} \sum_{k=1}^{n} \frac{1}{1+\frac{k}{n}} $$

And here you recognize a Riemann sum. So the limit is

$$\int_0^1 \frac{1}{1+x} dx$$

Edit : when something can be written as $\frac{1}{n} \sum \cdots$, a sum of Riemann may be hidden inside.

J. W. Tanner
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Tryss
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  • So what's the general form of a riemann sum? – N S Mar 07 '15 at 10:38
  • It's not always obvious you need to find a Riemann sum. The "general form" is $$\frac{1}{n} \sum_{k=0}^n f(\frac{k}{n})$$ But it's not always obvious that something can be written like that. As an exemple, can you find how to make appear a Riemann sum with this expression? $$ n^{-n^2} \prod_{k=1}^n (k+n)^{\frac{1}{n}}$$ – Tryss Mar 07 '15 at 11:54
  • By taking logarithm? @tryss – N S Mar 07 '15 at 12:17
  • Yes, you take the logarithm, and you make some manipulations to transform this nasty expression in a gentle Riemann sum. But would you think about Riemann sum if you stumbled onto this limit "in the wild"? – Tryss Mar 07 '15 at 12:19
  • No I wouldnt!! But thanks to you now I'd remember – N S Mar 07 '15 at 12:28
  • So is there any way I can solve that without Riemann? – N S Mar 07 '15 at 12:32
  • can you help me in this problem http://math.stackexchange.com/questions/1179407/let-theta-frac2-pi67-now-consider-the-matrix-beginpmatrix-cos-the – N S Mar 07 '15 at 12:32
  • http://prntscr.com/6dszmw How about this one? How to convert this into Riemann sum? – N S Mar 07 '15 at 13:33
  • I think it should be $ k/ n^2 $ – N S Mar 07 '15 at 13:41
  • Yes, I was going too fast ;) – Tryss Mar 07 '15 at 13:41
  • http://math.stackexchange.com/questions/1179473/riemann-sum-limits I've posted here. Can you help me? the k and n is having different powers. Or It can be done without Riemann Sum? – N S Mar 07 '15 at 13:43
4

More directly, multiplying by $1/n$ each term inside the parenthesis we get

$$\lim_{n\to\infty}\frac{1}{n+1}+\frac{1}{n+2}+\frac{1}{n+3}+\cdots+\frac{1}{2n}=$$ $$\lim_{n\to\infty}1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}+\frac{1}{n+1}+\cdots+\frac{1}{2n}-1-\frac{1}{2}-\frac{1}{3}-\cdots-\frac{1}{n}=\\\lim_{n\to\infty}\sum_{k=1}^{2n}\frac{1}{k}-\sum_{k=1}^n\frac{1}{k},$$ which, since the harmonic series grows logarithmically, is the same as $$\lim_{n\to\infty}\log(2n)-\log n=\log 2.$$

Vincenzo Oliva
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using digamma function we have $$\frac{1}{n+1} + \frac{1}{n+2} + \cdots + \frac{1}{2n}=\psi(n+1)-\psi(2n+1)$$ but then using the following asymptotic expansion $$\psi(x)= \ln(x) - \sum_{k=1}^\infty \frac{B_k}{k x^k}$$ with type two Bernoulli numbers $B_k$, we have \begin{align} \lim_{n \to \infty}\frac{1}{n+1} + \frac{1}{n+2} + \cdots + \frac{1}{2n}&=\lim_{n \to \infty}\Big(\psi(n+1)-\psi(2n+1)\Big)\\ &=\ln2 \end{align}

Math-fun
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Notice, we have $$\lim_{n \to +\infty}\frac{1}{n}\left({\frac{n}{n+1} + \frac{n}{n+2} + \cdots + \frac{n}{2n}}\right)$$

$$=\lim_{n\to +\infty}\sum_{r=1}^{n}\frac{1}{n}\left(\frac{n}{n+r} \right)$$ $$=\lim_{n\to +\infty}\sum_{r=1}^{n}\frac{1}{n}\left(\frac{1}{1+\frac{r}{n}} \right)$$

Let $\displaystyle \frac{r}{n}=x\implies \lim_{n\to \infty}\frac{1}{n}=dx\to 0$ $$ \text{lower limit of }\ x=\lim_{n\to \infty}\frac{r}{n}=\lim_{n\to \infty}\frac{1}{n}=0$$ $$ \text{upper limit of }\ x=\lim_{n\to \infty}\frac{r}{n}=\lim_{n\to \infty}\frac{n}{n}=1$$ Now, we have $$\lim_{n\to +\infty}\sum_{r=1}^{n}\frac{1}{n}\left(\frac{1}{1+\frac{r}{n}} \right)=\int_{0}^{1}\left(\frac{1}{1+x} \right)dx$$ $$=\left[\ln (1+x)\right]_{0}^{1}$$$$=\ln (2)-\ln(1)=\ln 2$$

juantheron
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Harish Chandra Rajpoot
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Since

$\begin{array}\\ \frac1{k} &> \int_k^{k+1} \frac{dx}{x} > \frac1{k+1}\\ \sum_{k=n+1}^{2n}\frac1{k} & > \sum_{k=n+1}^{2n}\int_k^{k+1} \frac{dx}{x} > \sum_{k=n+1}^{2n}\frac1{k+1}\\ or\\ \sum_{k=n+1}^{2n}\frac1{k} & > \int_{n+1}^{2n+1} \frac{dx}{x} > \sum_{k=n+1}^{2n}\frac1{k}-\frac1{n+1}+\frac1{2n+1}\\ \end{array} $

Since $\int_{n+1}^{2n+1} \frac{dx}{x} =\ln(2n+1)-\ln(n+1) =\ln(2-\frac1{n+1}) =\ln(2)-\ln(1-\frac1{2n+2}) $, if $d(n) =\ln(2)-\sum_{k=n+1}^{2n}\frac1{k} $, $d(n) < \ln(1-\frac1{2n+2}) < 0 $ and $d(n) >\ln(1-\frac1{2n+2})-\frac1{n+1}+\frac1{2n+1} $.

Setting $k=2n+1$ in $\frac1{k} > \int_k^{k+1} \frac{dx}{x} > \frac1{k+1} $ we get $\frac1{2n+1} > -\ln(1-\frac1{2n+1}) > \frac1{2n+2} $. Therefore $d(n) >-\frac1{2n+1}-\frac1{n+1}+\frac1{2n+1} =-\frac1{n+1} $. so $$-\frac1{n+1} < \ln(2)-\sum_{k=n+1}^{2n}\frac1{k} < 0. $$

marty cohen
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