Finding $\lim_{n\to \infty}\left((1+\frac{1}{n^2})(1+\frac{2^2}{n^2})^2\cdots(1+\frac{n^2}{n^2})^n\right)^{\frac{1}{n}}$ Calculate the following limit:
$$\lim_{n\to \infty}\left((1+\frac{1}{n^2})(1+\frac{2^2}{n^2})^2\cdots(1+\frac{n^2}{n^2})^n\right)^{\frac{1}{n}}$$

My attempt:

Let
$$y=\lim_{n\to \infty}\left((1+\frac{1}{n^2})(1+\frac{2^2}{n^2})^2\cdots(1+\frac{n^2}{n^2})^n\right)^{\frac{1}{n}}=\lim_{n\to \infty}\left((1+\frac{1}{n^2})^{\frac{1}{n}}(1+\frac{2^2}{n^2})^{\frac{2}{n}}\cdots(1+\frac{n^2}{n^2})^{\frac{n}{n}}\right)$$
Now, taking logarithm on both sides to get:  $$\log y=\lim_{n \to \infty}\sum_{k=1}^n\frac{k}{n}\log(1+(\frac{k}{n})^2)$$
But I am unable to convert it into integral form (riemann sum to integral ) as the expression $\frac{k}{n}$ is present instead of $\frac{1}{n}$. Can someone please help me in solving this question. Thanks in advance.

Considert the factors
$$\left (1+\frac{k^2}{n^2}\right )^k,\,\, n/2< k \le n.$$
Each of these is greater than
$$\left (1+\frac{(n/2)^2}{n^2}\right )^{n/2} = \left (\frac{5}{4}\right)^{n/2}.$$
There are more than $n/2$ such factors. Thus the product of these factors is at least $(5/4)^{n^2/4}.$ Now take the $n$th root to get $(5/4)^{n/4}\to \infty.$ Thus the limit of the original expression is $\infty.$
