Finding limit of the form $\lim_{n\rightarrow \infty}\bigg(1+\sum^{n}_{r=1}\frac{2}{\binom{n}{r}}\bigg)^n$

Question

$$\lim_{n\rightarrow \infty}\bigg(1+\sum^{n}_{r=1}\frac{2}{\binom{n}{r}}\bigg)^n$$

solution i try

$$\bigg(1+\frac{2}{n}\bigg)^n<\bigg(1+\sum^{n}_{r=1}\frac{2}{\binom{n}{r}}\bigg)<\bigg(1+\frac{2}{n}+\frac{2}{\binom{n}{2}}+\cdots +\frac{2}{\binom{n}{1}}+2\bigg)^n$$

I am struck here. did not know how to solve it . help me to do that problem . Thanks

The limit doesn't converge, because $1+\sum_{r=1}^n \frac{2}{\binom{n}{r}} > 3$ for all $n$. Did you mean to take the sum up to $n-1$? — Dark Malthorp, May 19 '18 at 02:55
Are you sure that the exponent is $n$ and not $\frac 1n$ instead ? — Claude Leibovici, May 19 '18 at 03:03
@Dark Malthorp can you explain me plz what will happen when we take sum up to $(n-1)$. — jacky, May 19 '18 at 05:29
@ Claudi Leibovici if exponent is $1/n$ then limit is $0$. Is iam right or not plz tell me. — jacky, May 19 '18 at 05:30
@jacky if you take the sum up to $(n-1)$ instead, it will at least not always be bounded below by 3, but looking at trancelocation's answer, the limit will still be divergent. — Dark Malthorp, May 19 '18 at 05:36
@Dark Malthorp answer is $e^2$ . i too have doubt about that question. Thank You — jacky, May 19 '18 at 05:38
Are you sure you copied down the formula correctly? What you have written certainly diverges to $+\infty$ — Dark Malthorp, May 19 '18 at 05:40
for these limits note that in general $\big(1+a/n+o(1/n)\big)^n=e^a+o(1)$ as $n\to+\infty$ — Pietro Majer, May 18 '19 at 08:26

score 2 · Accepted Answer · answered May 19 '18 at 03:22

You can find the limit of the sums of reciprocals of binomial coefficents on this MSE page:

$\lim_{n\rightarrow \infty}\sum^{n}_{r=\color{blue}{0}}\frac{1}{\binom{n}{r}} = 2$

So, for your limit we have $$1+\sum^{n}_{r=\color{blue}{1}}\frac{2}{\binom{n}{r}} \stackrel{n\rightarrow \infty}{\longrightarrow}3 \Rightarrow \lim_{n\rightarrow \infty}\bigg(1+\sum^{n}_{r=1}\frac{2}{\binom{n}{r}}\bigg)^n = +\infty$$

Finding limit of the form $\lim_{n\rightarrow \infty}\bigg(1+\sum^{n}_{r=1}\frac{2}{\binom{n}{r}}\bigg)^n$

1 Answers1