Given the geometric series $\sum n_0=\infty x^n$, find the sum of the series $\sum n_1=\infty n{x}^{n-1}$, where $\left|x\right|<1$.

Which of the following regions in the plane has an area equal to the limit $\underset{n\to \infty }{lim}\frac{1}{n}(1+\frac{1}{n}+\frac{1}{n^2}+\frac{1}{n^3}+\dots +\frac{1}{n^{n-1}})$? Do not evaluate the limit.

Suppose $a_n$ and $b_n$ are sequences with positive terms, and the series $\sum b_n$ is known to be convergent. Which of the following statements is true?

For which values of $x$ does the series $\sum _{n=1}^{\infty }\frac{x^n}{n}$ converge?

Use series to evaluate the limit: $\underset{x\to 0}{lim}\frac{1-\cos \left(4x\right)}{1+4x-e^{4x}}$.

Use series to evaluate the limit: $\underset{x\to 0}{lim}\frac{\left(\sin \right(3x)-3x+\frac{9}{2}{x}^3)}{x^5}$

Use series to evaluate the limit: $\underset{x\to 0}{lim}\frac{1-\cos (5x)}{1+5x-e^{5x}}$.

Use series to evaluate the limit: $\underset{x\to 0}{lim}\frac{1-\cos \left(2x\right)}{1+2x-e^{2x}}$.

Use the squeeze theorem to find the limit: $\underset{(x,y)\to (0,0)}{lim}xy-\sin \left(\frac{1}{x^2+y^2}\right)$. What is the value of this limit?