Calculus 2: test 2

Grading

This is a test for Calculus 2: course.

1. Evaluate the limit $$\lim\limits_{x\rightarrow1}\dfrac{\ln x}{x^{2}-1}.$$
2. Write an expression for the $n$th term of the sequence $$-\dfrac{1}{2},\dfrac{3}{4},-\dfrac{7}{8},\dfrac{15}{16},-\dfrac{31}{32},....$$
3. Find the sum of the series $$\sum_{n=0}^{\infty}\dfrac{1+2^{n}}{3^{n}}.$$
4. Apply the Integral Test to show that the $p$-series $$\sum\dfrac{1}{n^{1/3}}$$ diverges.
5. Test the following series for convergence (including absolute/conditional): $$\sum\dfrac{2n^{1/2}}{n^{2}-1}.$$
6. Test the following series for convergence (including absolute/conditional): $$\sum(-1)^{n}\dfrac{1}{\sqrt{n}}.$$
7. Test for convergence (including absolute/conditional): $$\sum\dfrac{(-3)^{n}}{n!}.$$

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1. Evaluate $$\lim\limits_{n\rightarrow\infty}\dfrac{(-1)^{n}}{n}$$ if it exists.
2. Write a formula for the $n$th term of the sequence $$-\dfrac{1}{2},\dfrac{3}{4},-\dfrac{7}{8},\dfrac{15}{16},-\dfrac{31}{32},....$$
3. Find the sum of the series $$\sum_{n=0}^{\infty}\dfrac{1+2^{n}}{3^{n}}.$$
4. Apply the Integral Test to show that the $p$-series $$\sum\dfrac{1}{n^{1/3}}$$ diverges.
5. Test the following series for convergence (including absolute/conditional): $$\sum\dfrac{2n^{1/2}}{n^{2}-1}.$$
6. Test the following series for convergence (including absolute/conditional): $$\sum(-1)^{2n}\dfrac{1}{n^{n}}.$$
7. Test for convergence (including absolute/conditional): $$\sum \dfrac{(-1)^{n}}{n^{1/2}}.$$
8. Find the radius and the interval of convergence of the series $$\sum\dfrac{(x-1)^{n}}{\sqrt{n}2^{n}}.$$