Infinite Sequences and Series (BC Only)

Assuming that a function $f(x)$ is positive, continuous, and decreasing for all $x \\geq M$ (where $M$ is some positive number), determine whether the improper integral $\int_{M}^{\\infty} f(x) , dx$ converges or diverges if it is known that $\lim_{x \\to \\infty}(x^2 f(x)) = L > 0$.

Considering the alternating series $\sum_{n=1}^{\infty}(-1)^{n+1} \frac{n^{k}}{e^n}$, where k is an integer, what is the critical value of k at which point the Integral Test cannot determine convergence or divergence?

For which range of values does a power function diverge when subjected to an analysis using both Comparison and Limit Comparison Tests in conjunction with Integral Test?

Considering a series $\sum_{n=1}^{\infty} \frac{\sin(n)}{n^{p}}$, what value of $p$ ensures that applying the integral test reveals convergence?

How would replacing $f(x) = \frac{e^{-x}}{x}$ with $g(x) = \frac{e^{-x}}{(x+5)^2}$ influence the outcome when employing an integral test on these functions?

Which integral represents the length of an arc defined by the polar equation $r(\theta) = \cos(\theta)$ from $\theta = 0$ to $\theta = \frac{\pi}{4}$?

For which of the following series does applying the integral test require evaluating an improper integral of a rational function that necessitates partial fraction decomposition?

For which value of $p$ does the infinite series $\sum_{n=2}^{\infty} \frac{1}{n(\log n)^p}$ converge according to the Integral Test?

What is the formula to find the area of a sector in polar coordinates with radius r and angle θ measured in radians?

If a series $\sum_{n=1}^{\infty}$ does not decrease everywhere and is defined but is not always positive, how would this impact the application of the Integral Test?