Sequences and series of functions

Determine the pointwise limit on $[0,+\infty )$ of the sequence of functions

$$f_n(x)=\frac{1}{1+x^n}.$$

Solution available to those who have unlocked the subject.

Determine the pointwise limit on $[0,+\infty )$ of the sequence of functions

$$f_n(x)=n\ln \left(1+\frac{x}{n}\right).$$

Solution available to those who have unlocked the subject.

Study the pointwise and uniform convergence on $[0,1]$ of the sequence of functions

$$f_n(x)=x{e}^{-x/n}.$$

Solution available to those who have unlocked the subject.

Study the pointwise and uniform convergence on $[0,+\infty )$ of the sequence of functions

$$f_n(x)=x^2{e}^{-nx}.$$

Solution available to those who have unlocked the subject.

Study the pointwise and uniform convergence of the sequence of functions

$$f_n(x)=\frac{nx}{1+n^2{x}^2}$$

on $\mathbb{R}$ and on the half-lines $[\delta ,+\infty )$ with $\delta >0$.

Solution available to those who have unlocked the subject.

Determine the pointwise limit on $[0,1]$ of the sequence of functions

$$f_n(x)=x^{1/n}$$

and establish whether the convergence is uniform.

Solution available to those who have unlocked the subject.

Verify that the sequence of functions

$$f_n(x)=\frac{\sin (nx)}{\sqrt{n}}$$

converges uniformly on $\mathbb{R}$ and that the sequence of the derivatives evaluated at $0$ diverges.

Solution available to those who have unlocked the subject.

Compute

$$\underset{n\to +\infty }{\lim }{\int }_0^1{x}^n{e}^{x}dx.$$

Solution available to those who have unlocked the subject.

Compute

$$\underset{n\to +\infty }{\lim }{\int }_1^2\frac{e^{x/n}}{x}dx.$$

Solution available to those who have unlocked the subject.

Compute

$$\underset{n\to +\infty }{\lim }{\int }_0^1{n}^{2}x(1-x{)}^{n}dx$$

and deduce that the convergence of the sequence $f_n(x)=n^{2}x(1-x{)}^n$ is not uniform on $[0,1]$.

Solution available to those who have unlocked the subject.

Study the total convergence on $\mathbb{R}$ of the series of functions

$$\sum _{n=1}^{\infty }\frac{\cos (nx)}{n^2+1}.$$

Solution available to those who have unlocked the subject.

Study the total convergence on $[0,+\infty )$ of the series of functions

$$\sum _{n=1}^{\infty }{x}^2{e}^{-nx}.$$

Solution available to those who have unlocked the subject.

Prove that the series of functions

$$\sum _{n=1}^{\infty }\frac{(-1{)}^n}{2n+x^2}$$

converges uniformly on $\mathbb{R}$ but not totally.

Solution available to those who have unlocked the subject.

Prove that the function

$$f(x)=\sum _{n=1}^{\infty }\frac{\sin (nx)}{n^4}$$

is of class $C^1$ on $\mathbb{R}$ and compute its derivative.

Solution available to those who have unlocked the subject.