Calculus 2

Given the recursive sequence $c_0 = 2$ and $c_{n+1} = c_n + 4$, find the first five terms of the sequence.

Determine if the sequence $a_n = \frac{n-1}{n!}$ converges or diverges as $n \to \infty$.

Determine if the sequence $a_n = 1 + \frac{(-1)^n}{n^2}$ converges or diverges as $n \to \infty$.

Determine if the sequence $a_n = \sqrt[3]{\frac{n}{n+1}}$ converges or diverges as $n \to \infty$.

Explore whether the infinite series from n equals 1 to infinity of $\frac{1}{n^2}$ converges or diverges using the integral test.

Identify which convergence test to use for a geometric series involving terms like $\frac{2^n}{5^n}$.

For a series represented with a corresponding function over an interval, use the integral test to determine convergence.

Determine which of the given series might be suitable for an alternating series test, especially those containing terms like $(-1)^n$.

Apply divergence test to a series where the sequence does not converge to zero, such as one involving logarithmic terms.

For series that resemble quotient forms, like $\frac{n}{n^3 + 1}$, determine convergence using the comparison or limit comparison test.

Attempt comparing series with non-standard terms using the limit comparison test.

Utilize the root test for series with terms that include powers like $(\text{something})^n$.

Apply the ratio test to series involving factorial terms and powers, such as those with $n!$ or similar structures.

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

Try using the integral test on your own for the series $\sum_{n=1}^{\infty} e^n$ and determine if it converges or diverges.

If you look at the improper interval and the improper integral converges or diverges, whatever it does, the same is true of the series.

Calculate whether the series $\sum_{n=1}^{\infty} \frac{1}{n^2}$ is convergent using the integral test, and estimate its sum.

Using the comparison test, determine if the series $\displaystyle \sum_{n=1}^{\infty} \frac{1}{2^n + 1}$ is convergent by comparing it to the series $\displaystyle \sum_{n=1}^{\infty} \frac{1}{2^n}$.

Determine whether the series $\displaystyle \sum_{n=1}^{\infty} \frac{5}{2n^2 + 4n + 3}$ is convergent using the comparison test.

Determine whether the series represented by the function is convergent or divergent using the integral test for the function: $\displaystyle \int_{1}^{\infty} \frac{1}{2^x} \, dx$