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 exercise, find a geometric power series for the function, centered at $0$, by the technique shown in the previous examples.

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

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

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

Use a graph to show that

$$\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}>\int_1^{\infty} \frac{1}{\sqrt{x}} d x.$$

What can you conclude about the convergence or divergence of the series? Explain.

In this exercise, find the sum of the convergent series.

$$\displaystyle\sum_{n=1}^{\infty}(\sin 1)^n$$

In this exercise, verify the sum. Then use a graphing utility to approximate the sum with an error of less than $0.0001$.

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

Prove that $\arctan x+\arctan y=\arctan \dfrac{x+y}{1-x y}$ for $x y \neq 1$ provided the value of the left side of the equation is between $-\pi / 2$ and $\pi / 2$.

In this task, determine whether the series converges conditionally or absolutely, or diverges.

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

In this task, determine the convergence or divergence of the sequence with the given $n$th term. If the sequence converges, find its limit.

$$a_n=\dfrac{3 n^2-n+4}{2 n^2+1}$$

Review the results of the pervious exercises. Explain why careful analysis is required to determine the convergence or divergence of a series and why only considering the magnitudes of the terms of a series could be misleading.

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

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

In this exercise, determine the degree of the Maclaurin polynomial required for the error in the approximation of the function at the indicated value of $\boldsymbol{x}$ to be less than $0.001$.

$$\sin (0.3)$$

In this exercise, find the intervals of convergence of $\int f(x) d x$. Include a check for convergence at the endpoints of the interval.

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

In this exercise, find the intervals of convergence of $f^{\prime \prime}(x)$. Include a check for convergence at the endpoints of the interval.

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