Find the interval of convergence.

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

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

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

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

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

Determine the convergence or divergence of the integrals.

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

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

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

Test for convergence.

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

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

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

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.

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

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

What are significant digits? How can you tell whether zeros are significant?

Find a power series representation for the function and determine the interval of convergence. \

$$f(x)=\dfrac{1}{1+9 x^2}$$

Find the series’ radius and interval of convergence. For what values of x does the series converge?

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

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

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