MHB Evaluate Product: Limit of Product Involving Tangents

[lfdahl]

Evaluate the product:

\[\lim_{n\to \infty}\prod_{k=1}^{n}\left(1+\tan \left(\frac{1}{n+k}\right)\right), \;\;\; n,k \in \Bbb{N}.\]

 

[lfdahl]

Hint:

Spoiler

\[\lim_{x \rightarrow 0}\frac{\tan x}{x} = 1\]

 

[lfdahl]

Suggested solution:

Spoiler

Let the limit be $L$.We have:

\[\lim_{n\rightarrow \infty }\frac{\tan\left ( \frac{1}{n+k} \right )}{\frac{1}{n+k}} =\lim_{n\rightarrow \infty }(n+k)\tan\left ( \frac{1}{n+k} \right ) = 1,\;\;\; 1 \le k \le n.\]

Hence:

\[L = \lim_{n \rightarrow \infty }\prod_{k=1}^{n}\left ( 1+ \frac{1}{n+k}\right ) = \lim_{n\rightarrow \infty } \frac{2n+1}{n+1} =2.\]