Numerical series

Compute the sum of the series

$$\sum _{n=0}^{\infty }{\left(\frac{2}{5}\right)}^n.$$

Solution available to those who have unlocked the subject.

Compute the sum of the series

$$\sum _{n=1}^{\infty }\frac{3^{n-1}}{4^n}.$$

Solution available to those who have unlocked the subject.

Compute the sum of the series

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

Solution available to those who have unlocked the subject.

Compute the sum of the series

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

Solution available to those who have unlocked the subject.

Study the behavior of the series

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

Solution available to those who have unlocked the subject.

Study the behavior of the series

$$\sum _{n=1}^{\infty }\frac{\sqrt{n+1}-\sqrt{n}}{n}.$$

Solution available to those who have unlocked the subject.

Study the behavior of the series

$$\sum _{n=1}^{\infty }\left(e^{\frac{1}{n}}-1\right).$$

Solution available to those who have unlocked the subject.

Study the behavior of the series

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

Solution available to those who have unlocked the subject.

Study the behavior of the series

$$\sum _{n=1}^{\infty }\frac{(2n)!}{(n!{)}^2{5}^n}.$$

Solution available to those who have unlocked the subject.

Study the behavior of the series

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

Solution available to those who have unlocked the subject.

Study the absolute convergence and the convergence of the series

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

Solution available to those who have unlocked the subject.

Study the absolute convergence and the convergence of the series

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

Solution available to those who have unlocked the subject.

Determine for which $x\in \mathbb{R}$ the series

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

converges and compute its sum.

Solution available to those who have unlocked the subject.

Determine the set of convergence of the power series

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

Solution available to those who have unlocked the subject.