In this exercise, determine the convergence or divergence of the series.

$$\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^n \sqrt{n}}{n+1}$$

Find the series' radius and interval of convergence.

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

In this task, determine the convergence or divergence of the series.

$$\displaystyle\sum_{n=1}^{\infty} \frac{n}{\sqrt{n^2+1}}$$

In this task, determine the convergence or divergence of the series.

$$3\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^{0.95}}$$

In this exercise, find the radius of convergence of the power series.

$$\displaystyle\sum_{n=1}^{\infty} \frac{(2 x)^n}{n^2}$$

Determine the convergence or divergence of the series. $\sum_{n=1}^{\infty}\left(\frac{1}{2 n(n+1)}\right)$

Test for convergence.

$$\sum_{n=1}^{\infty} \frac{\sqrt{n}+n+n^{3 / 2}}{\sqrt{n}+n+n^{5 / 2}+n^3}$$

Use the Ratio Test to determine the convergence or divergence of the series. $\sum_{n=1}^{\infty} \frac{n^{3}}{4^{n}}$

Determine the convergence or divergence of the integrals.

$$\int_{-1}^{\infty} \frac{\tan ^{-1} x}{(2+x)^3} d x$$

Consider the Fourier sine series of each of the following functions. In this exercise do not compute the coefficients but use the general convergence theorems to discuss the convergence of each of the series in the pointwise, uniform, and $L^2$ senses. (a) $f (x) = x^3$ on (0, l). (b) $f (x) = lx − x^2$ on (0, l). (c) $f (x) = x−2$ on (0, l).

Find the interval of convergence.

$$\sum \frac{k}{2 k+1} x^{2 k+1}$$

Solve the absolute value inequality to find the interval of convergence.

$$\left| \dfrac{x+1}{2} \right| < 1$$

In this exercise, determine the convergence or divergence of the series.

$$\displaystyle\sum_{n=0}^{\infty} \dfrac{(-1)^n}{(2 n+1) !}$$

Find the interval of convergence.

$$\sum(-1)^k \frac{k^2}{(k+1) !}(x+3)^k$$