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

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

In this exercise, use the power series

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

to determine a power series, centered at $0$, for the function. Identify the interval of convergence.

$$f(x)=\arctan 2 x$$

In this exercise, find the Maclaurin series for the function. (Use the table of power series for elementary functions.)

$$f(x)=\cos x^{3 / 2}$$

In this exercise, use the power series

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

to determine a power series, centered at $0$, for the function. Identify the interval of convergence.

$$h(x)=\dfrac{1}{4 x^2+1}$$

In this exercise, find the $n$th Taylor polynomial centered at $c$.

$f(x)=\frac{1}{x}, \quad n=4, \quad c=1$

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

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

In this exercise, use the Integral Test to determine the convergence or divergence of the $p$-series.

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

In this exercise, use the Limit Comparison Test to determine the convergence or divergence of the series.

$$\displaystyle\sum_{n=1}^{\infty} \frac{n^{k-1}}{n^k+1}, \quad k>2$$

In this task, write the next two apparent terms of the sequence. Describe the pattern you used to find these terms.

$$2,5,8,11, \ldots$$

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

$$\displaystyle\sum_{n=0}^{\infty} \frac{4^n}{n !}$$

In this exercise, find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.)

$$\displaystyle\sum_{n=1}^{\infty} \frac{(x-3)^{n-1}}{3^{n-1}}$$

In this task, verify that the infinite series converges.

$$\displaystyle\sum_{n=0}^{\infty} 2\left(\frac{3}{4}\right)^n$$

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

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

In this exercise, use the Limit Comparison Test to determine the convergence or divergence of the series.

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