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

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

Consider the formula

$$\frac{1}{x-1}=1+x+x^2+x^3+\cdots$$

Given $x=-1$ and $x=2$, can you conclude that either of the following statements is true? Explain your reasoning.

$$-1=1+2+4+8+\cdots$$

Consider the formula

$$\frac{1}{x-1}=1+x+x^2+x^3+\cdots$$

Given $x=-1$ and $x=2$, can you conclude that either of the following statements is true? Explain your reasoning.

$$\dfrac{1}{2}=1-1+1-1+\cdots$$

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

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

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

$$\displaystyle\sum_{n=1}^{\infty}(-1)^{n+1} \frac{1}{5^n n}$$

In this task, test for convergence or divergence and identify the test used.

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

In this exercise, find the values of $\boldsymbol{x}$ for which the series converges.

$$\displaystyle\sum_{n=0}^{\infty}\left(\frac{x+1}{4}\right)^n$$

In this task, determine whether the sequence with the given $n$th term is monotonic. Discuss the boundedness of the sequence. Use a graphing utility to confirm your results.

$$a_n=\left(-\dfrac{2}{3}\right)^n$$

You add a finite number of terms to a convergent series. Will the new series still converge? Explain your reasoning.

You delete a finite number of terms from a divergent series. Will the new series still diverge? Explain your reasoning.

In this task, test for convergence or divergence and identify the test used.

$$\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^{n+1} 4}{3 n^2-1}$$

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

$$\displaystyle\sum_{n=1}^{\infty}\left(1+\frac{1}{n}\right)^n$$

In this task, determine whether the sequence with the given $n$th term is monotonic. Discuss the boundedness of the sequence. Use a graphing utility to confirm your results.

$$a_n=(-1)^n\left(\dfrac{1}{n}\right)$$

In this exercise, find the values of $\boldsymbol{x}$ for which the series converges.

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