Power series and Taylor series

Determine the radius and the set of convergence of the series

$$\sum _{n=1}^{\infty }\frac{x^n}{n^2{2}^n}.$$

Solution available to those who have unlocked the subject.

Determine the radius and the set of convergence of the series

$$\sum _{n=1}^{\infty }n{3}^n{x}^n.$$

Solution available to those who have unlocked the subject.

Determine the radius and the set of convergence of the series

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

Solution available to those who have unlocked the subject.

Determine the radius and the set of convergence of the series

$$\sum _{n=0}^{\infty }\frac{x^n}{5^{n+(-1{)}^n}}.$$

Solution available to those who have unlocked the subject.

Determine the set of convergence of the series

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

Solution available to those who have unlocked the subject.

Determine the set of convergence of the series

$$\sum _{n=1}^{\infty }\frac{(x+2{)}^n}{n^2{3}^n}.$$

Solution available to those who have unlocked the subject.

Determine the radius and the set of convergence of the series

$$\sum _{n=1}^{\infty }\frac{x^{2n}}{n25^n}.$$

Solution available to those who have unlocked the subject.

Determine the radius and the set of convergence of the series

$$\sum _{n=1}^{\infty }\frac{n{x}^{2n+1}}{4^n}.$$

Solution available to those who have unlocked the subject.

Determine the Maclaurin expansion of

$$f(x)=x{e}^{-2x}$$

and its region of validity.

Solution available to those who have unlocked the subject.

Determine the Maclaurin expansion of

$$f(x)=\frac{1}{5+x}$$

and its region of validity.

Solution available to those who have unlocked the subject.

Determine the Maclaurin expansion of the integral function

$$F(x)={\int }_0^x\frac{\sin t}{t}dt,$$

where the integrand is extended with the value $1$ at $t=0$.

Solution available to those who have unlocked the subject.

Compute the sum of the series

$$\sum _{n=1}^{\infty }\frac{n}{5^n}.$$

Solution available to those who have unlocked the subject.

Compute the sum of the series

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

Solution available to those who have unlocked the subject.

Compute, using Maclaurin expansions, the limit

$$\underset{x\to 0}{\lim }\frac{e^{x^2}-1-x^2}{x^4}.$$

Solution available to those who have unlocked the subject.