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

Answer 1

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 $$1+\sum^{n}_{r=\color{blue}{1}}\frac{2}{\binom{n}{r}} \stackrel{n\rightarrow \infty}{\longrightarrow}3 \Rightarrow \lim_{n\rightarrow \infty}\bigg(1+\sum^{n}_{r=1}\frac{2}{\binom{n}{r}}\bigg)^n = +\infty$$