Calculus 2

Use the direct comparison test to determine if the series $\sum_{n=1}^{\infty} \frac{2^n}{7^n + 8}$ converges or diverges.

Determine the convergence or divergence of the series $\sum_{n=1}^{\infty} \frac{\ln(n)}{n}$.

Determine the convergence or divergence of the series {n=1}^{\infty} \frac{n+5}{n^2}.

Determine the convergence or divergence of the series $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n^4 + 5}}$.

Use the limit comparison test to determine if the series $\frac{n^2}{n^5 + 8}$ converges or diverges.

Use the limit comparison test to see if the series $\frac{1}{\sqrt{n^2 + 2}}$ converges or diverges.

Determine if the series $\frac{1}{3^n + 5}$ converges or diverges using the limit comparison test.

Suppose that the two series $a_n$ (the series we care about) and $b_n$ (the series we will use for comparison) have positive terms. If the series $b_n$ is convergent and the terms $a_n \leq b_n$ for all $n$, then the series $a_n$ converges.

Using the direct comparison test, determine whether a series converges or diverges when one series is bounded by another, given that both sequences are positive.

Suppose the integral from 2 to infinity of $f(x)$ converges to a finite value $L$. What can be said about the integral from 2 to infinity of $g(x)$, given that $f(x) > g(x) > 0$? Does it converge or diverge?

Consider the integral from 1 up to infinity of rac{x-2}{x^3+1}. Determine if this integral converges or diverges.

For our first actual example, we're going to pretend this random differential equation with initial conditions that I just got from my textbook yields a solution that's very important to an engineer or scientist.

The question is: If we have a single positive charge in space, then the electric field at some point is inversely proportional to the distance from that charge squared. Now, if we double that distance from the origin on that same axis, what kind of behavior will we find for the electric field?

Find the Taylor polynomial of degree n at x = C.

Using the Maclaurin series for $\cos(x)$, rewrite the series to accommodate $\cos(2x)$, and simplify the expression as necessary.

Approximate the sin function using a Taylor polynomial.

Using the ratio test, determine the intervals of convergence for the power series 28 29. Answer: The interval of convergence is $2 < x < 4$.

Determine the interval of convergence for the power series $\sum_{n=0}^{\infty} \frac{x^n}{n!}$ using the ratio test.

Determine the interval of convergence for the power series: $\sum \frac{1}{n} (x - 2)^n$.

Find the interval of convergence for the power series with $c_n = \frac{1}{n!}$ and $(x - 2)^n$.