As the title suggests, how can I find the answer of this limit?
I do not know exactly about what kind of this problem belongs to, maybe estimation or evaluation.
$$\lim\limits_{n \to \infty} n \left[(\dfrac 1 {\pi} \sum_{k=1}^n \sin\dfrac{\pi} {\sqrt{n^2+k}})^n - \dfrac 1 {\sqrt[4]{e}}\right]$$
Minutes passed; I find the core of the limit (in the parentheses) equals to the following formula $$1 - \dfrac 1 {4n} + (\dfrac {\pi^2} 6 - \dfrac 1 8)\dfrac 1 {n^2} + o(\dfrac 1 {n^2})$$
Could someone give hints on how I would go about the next step?
Thank you very much.