Asymptotic behavior of Wallis integrals

Let us denote with $W_n$ the Wallis integral, that is:

$$W_n=\int_0^{\pi/2} \sin^n x \, {\rm d}x$$

Prove that $W_n \sim \sqrt{\frac{\pi}{2n}}$.

Solution

A limit

Evaluate the limit:

$$\lim_{ n \rightarrow +\infty} \left[ \frac{\arctan 1}{n+1} + \frac{\arctan 2}{n+2}+\cdots+\frac{\arctan n }{n+n} \right]$$

Solution

Limit of nested integrals

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that $f(x) \geq 0$ forall $x$ and $\displaystyle \int_{-\infty}^{\infty} f(x)\, {\rm d}x=1$. For $r \geq 0$ , define:

$$I_n(r)= \idotsint \limits_{x_1^2+x_2^2+\cdots+x_n^2 \leq r} f(x_1) f(x_2) \cdots f(x_n) \;{\rm d}(x_1, x_2, \dots, x_n)$$

Evaluate the limit $\lim I_n(r)$ for a fixed $r$.

Solution

A limit from derivative

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function at $x_0=0$ and $f(0)=0$ and let $k \in \mathbb{N}$. Evaluate the limit:

$$\lim_{x\rightarrow 0}\frac{1}{x}\left [ f(x)+ f\left ( \frac{x}{2} \right )+ \cdots+ f\left ( \frac{x}{k-1} \right )+ f\left ( \frac{x}{k} \right ) \right ]$$

Solution

An interesting limit (2)

Evaluate the limit:

$$\lim_{ n \rightarrow +\infty} \frac{1}{\Gamma(n)} \int_0^n x^{n-1}e^{-x}\, {\rm d}x$$

Solution

An interesting limit (1)

Evluate the limit:

$$\lim_{n \to +\infty} \left[ {e}^{-n} \sum_{r=0}^{n} \frac{ n^r }{ r! } \right] = \frac{1}{2}$$

Solution

Limit and number theory

Let $\varphi $ denote the Euler function. Evaluate the limit:

$$\lim_{n \rightarrow +\infty} \frac{1}{n^2} \sum_{k \leq n} \varphi(k)$$

Solution

A limit

Evaluate the limit:

$$\lim_{n \to +\infty} \int_{0}^1 \frac{n+1}{2^{n+1}} \left(\frac{(t+1)^{n+1}-(1-t)^{n+1}}{t}\right) \,{\rm d}t$$

Solution: