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

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

In thsi exercise, find the third-degree Taylor polynomial centered at $c$.

$$f(x)=e^{-x / 2}, \quad c=0$$

In this exercise, determine the values of $\boldsymbol{x}$ for which the function can be replaced by the Taylor polynomial if the error cannot exceed $0.001$.

$$f(x)=e^x \approx 1+x+\dfrac{x^2}{2 !}+\dfrac{x^3}{3 !}, \quad x<0$$

In this exercise, use the result of Exercise 51 to determine the convergence or divergence of the series.

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

In this exercise, use a power series to approximate the value of the integral with an error of less than $0.0001$.

$$\displaystyle\int_{0.1}^{0.3} \sqrt{1+x^3} d x$$

In this exercise, find the sum of the convergent series by using a well-known function. Identify the function and explain how you obtained the sum.

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

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

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

In this exercise, determine the convergence or divergence of the series using any appropriate test from this chapter. Identify the test used.

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

In this exercise, write its sum as the ratio of two integers.

$$0.2\overline{15}$$

In this exercise, write the repeating decimal as a geometric series.

$$0.2\overline{15}$$

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 $x$ to be less than $0.0001$. Use a computer algebra system to obtain and evaluate the required derivatives.

$f(x)=e^{-x}$, approximate $f(1)$.

In this task, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $0<a_{n+10} \leq b_n$ and $\displaystyle\sum_{n=1}^{\infty} b_n$ converges, then $\displaystyle\sum_{n=1}^{\infty} a_n$ converges.

In this exercise, use the result of Exercise 51 to determine the convergence or divergence of the series.

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

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

$$\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{1.5}}$$