I have the following limit:
$$\lim_{n \to \infty} \int_{- \infty}^{\infty} \frac{\cos x}{1+x^{2n}}~dx$$
Based on experimentation on WolframAlpha, it would seem that the value converges to a number close to 1.68294... So I have two questions: 1) How would I show this sequence converges at all and 2) is the value that the sequence converges to expressible in terms of a series or other fundamental constants etc., or is it just an irrational value that must be taken for what it is?
$\int_{-1}^{1} \frac {\cos x} {1+x^{2n}} \, dx \to \int_{-1}^{1} \cos x \, dx$ , $\int_{-\infty} ^{-1} \frac {\cos x} {1+x^{2n}} \, dx \to 0$ and $\int_{1} ^{\infty } \frac {\cos x} {1+x^{2n}} \, dx \to 0$ so the limit is $\int_{-1}^{1} \cos x \, dx=\sin 1 - \sin (-1)=2\sin 1$. For justifying the second and third limits use the fact that $\frac 1 {1+x^{2n}} \leq \frac 1 {1+x^{2n}}$ and apply DCT.