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

Given a curve $C$ and a contour map of a function $f$ whose gradient is continuous, which of the following statements is true about the direction of the gradient vector of $f$ at a point on $C$?

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?

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 $\left|x\right|<1$.

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