Unit 6 test: Infinite Sequences and Series

Find $\displaystyle\lim_{n\to\infty} \frac{4n^3 - n}{6n^3 + 5n^2}$.

What can be concluded about $\displaystyle\sum_{n=1}^{\infty} \frac{3n - 1}{5n + 2}$?

Find the sum of $\displaystyle\sum_{n=1}^{\infty} 6\left(\frac{3}{7}\right)^n$.

The integral test is applied to $\displaystyle\sum_{n=1}^{\infty} \frac{1}{(2n + 1)^2}$. Which is the correct conclusion?

Which of the following series converges?

To apply the limit comparison test to $\displaystyle\sum_{n=1}^{\infty} \frac{n^2 + 3}{2n^4 - n}$, it is compared with $\displaystyle\sum \frac{1}{n^2}$. Find $\displaystyle\lim_{n\to\infty} \frac{a_n}{b_n}$.

The series $\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^2}$ is approximated by $S_5$, the sum of its first five terms. Using the alternating series error bound, find the best upper bound on the error.

For $\displaystyle\sum_{n=1}^{\infty} \frac{n^3\, 2^n}{5^n}$, find the ratio test limit $L = \displaystyle\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|$.

Classify $\displaystyle\sum_{n=2}^{\infty} \frac{(-1)^n}{n \ln n}$.

A function $f$ has $f(3) = 1$, $f'(3) = -2$, $f''(3) = 5$ and $f'''(3) = 6$. Write the third-degree Taylor polynomial for $f$ about $x = 3$.

A function $g$ has derivatives of all orders and satisfies $|g^{(5)}(x)| \le 12$ on $[0, 0.5]$, and its fourth-degree Maclaurin polynomial is used to approximate $g(0.5)$. Find the Lagrange error bound.

Find the interval of convergence of $\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^n (x + 1)^n}{n\,2^n}$.

What is the coefficient of $x^3$ in the Maclaurin series for $e^{2x}$?

Find the exact sum of $\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n\, 2^n}$.

Use series to find $\displaystyle\lim_{x\to 0} \frac{x - \arctan x}{x^3}$.