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

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.

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

Find the interval of convergence.

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

Use the Integral Test to determine the convergence or divergence of the p-series. $\sum_{n=1}^{\infty} \frac{1}{n^{0.9}}$

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

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

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

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

Find the radius of convergence of the power series. $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}(x-5)^{n}}{n 5^{n}}$

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

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

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

$$\displaystyle\sum_{n=1}^{\infty} \frac{n+10}{10 n+1}$$

Test for convergence.

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

Find the interval of convergence.

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

Determine the convergence or divergence of the integrals.

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

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

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