Sequences and Series — Quiz 3

Questions in this quiz (9)

1. Determine whether or not the series $$\sum_{n=1}^{\infty}\left(1+\frac{2}{n}\right)^{-n}$$ converges.

2. Find the interval of convergence for the power series $$1-\frac{x}{2}+\frac{x^2}{2\cdot4}-\frac{x^3}{3\cdot8}+\cdots$$.

3. If $$P(x)=1+2(x-2)-3(x-2)^2$$ is a Taylor polynomial for $$f(x)$$, write a third-degree Taylor polynomial for $$m(x)=\int_2^x P(t)\,dt$$.

4. Does the series $$\sum_{n=1}^{\infty}\frac{(-1)^n}{(2n+1)!}$$ converge absolutely, conditionally, or diverge?

5. Find the value of $$a$$ so that the radius of convergence for the series $$\sum_{n=1}^{\infty}\frac{(a^3)^n(x-2)^n}{n^2}$$ is $$\frac13$$ and $$a>0$$.

6. If $$g(x)=e^{\cos x}$$, use a fourth-degree Maclaurin polynomial to approximate $$g''(0.2)$$.

7. Find the sum of the series $$2-\frac23+\frac29-\frac{2}{27}+\cdots$$.

   • $$\frac32$$
   • $$\frac43$$
   • $$\frac65$$
   • $$\frac54$$

8. Find the sum of the series $$\sum_{n=1}^{\infty}(-1)^n e^{-2n}$$.

   • $$\frac{-e^{-2}}{1-e^{-2}}$$
   • $$\frac{e^{-2}}{1-e^{-2}}$$
   • $$\frac{-e^{-2}}{1+e^{-2}}$$
   • $$\frac{1}{1-e^{-2}}$$

9. Find the sum of the series $$\sum_{n=1}^{\infty}\left(\frac{1}{n+1}-\frac{1}{n+4}\right)$$.

   • $$\frac56$$
   • $$\frac{13}{12}$$
   • $$\frac{7}{12}$$
   • $$\frac{11}{12}$$