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.

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

Find the interval of convergence.

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

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

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

Test for convergence.

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

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

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

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

$$\displaystyle\sum_{n=2}^{\infty} \frac{1}{n(\ln n)^3}$$

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.

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

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

$$\displaystyle\sum_{n=0}^{\infty}(-1)^n \frac{x^n}{n+1}$$

In this task, use the Ratio Test to determine the convergence or divergence of the series.

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

Test the given series for convergence.

$$\sum_{k=2}^{\infty} \frac{1}{k\ ln\ k}$$

Find the Taylor series about the indicated center and determine the interval of convergence. $f(x)=\frac{1}{x}, c=1$

If $k$ is a positive integer, find the radius of convergence of the series

$$\sum_{n=0}^2 \frac{(n !)^k}{(k n) !} x^n$$