$\lim b_n=\frac{n^n}{(n+1)(n+2)\dots(n+n)}.$ $$b_n=\frac{n^n}{(n+1)(n+2)\dots(n+n)}.$$
Now, there is this theorem for sequences that if $\lim_{n\to ∞}⁡ a_{n+1} /a_n =l$, $|l|<1$ then $\lim_{n\to ∞}⁡ a_n=0$.
so, $\lim_{n\to ∞}⁡ b_{n+1} /b_n =e/4$ which is less than $1$, so $\lim_{n\to ∞}⁡ b_n$ should be equal to zero.
But if I calculate the limit of $b_n$ as, $b_n=(n\cdot n\cdot n\cdots n)/((n+1)(n+2)...(n+n))$ I get $\lim_{n\to ∞}⁡ b_n=1/2$.
Something is definitely going wrong. Can someone point out my mistake,Please.

I assume your argument looks like this:
$$b_n=\frac{n}{n+1} \frac{n}{n+2} ... \frac{n}{n+n}$$
Taking the limit as $n\to\infty$, each factor goes to $1$, eg. $\frac n {n+1}\to 1$, except the last one, since $\frac n {n+n}=\frac 1 2$. Thus the limit is $\frac 1 2$.
Well, what about the second-to-last factor? $\frac{n}{n+n-1}\to\frac 1 2$ as well. So is the limit $\frac 1 4$?
The basic problem is that the rule
$$\lim_{n\to\infty} a_n b_n = (\lim_{n\to\infty}a_n)(\lim_{n\to\infty}b_n)$$
Only applies when the number of factors doesn't depend on $n$.
