Section 11.3

Key Problems: 3, 5, 8, 13, 17, 23, 26, 32, 39

Problem 3

Use the Integral Test to determine whether the series is convergent or divergent.

$$\sum _{n=1}^{\infty }{n}^{-3}$$

Problem 5

$$\sum _{n=1}^{\infty }\frac{2}{5n-1}$$

Problem 8

$$\sum _{n=1}^{\infty }{n}^2{e}^{-n^3}$$

Problem 13

Determine whether the series is convergent or divergent.

$$1+\frac{1}{8}+\frac{1}{27}+\frac{1}{64}+\frac{1}{125}+\dots$$

Problem 17

$$\sum _{n=1}^{\infty }\frac{\sqrt{n}+4}{n^2}$$

Problem 23

$$\sum _{n=2}^{\infty }\frac{1}{n\ln n}$$

Problem 26

$$\sum _{k=1}^{\infty }k{e}^{-k^2}$$

Problem 32

Find the values of $p$ for which the series is convergent.

$$\sum _{n=3}^n\frac{1}{n\ln n[\ln (\ln n){]}^p}$$

Problem 39

$$\zeta (2)=\sum _{n=1}^{\infty }\frac{1}{n^2}=\frac{{\pi }^2}{6}$$

(a) Use the sum of the first $10$ terms to estimate the sum of the series ${\sum }_{n=1}^{\infty }1/n^2$. How good is this estimate?

(b) Improve this estimate using (3) with $n=10$.

(c) Compare your estimate in part (b) with the exact value of $\zeta (2)$¹.

(d) Find a value of $n$ that will ensure that the error in the approximation $s\approx s_n$ is less than $0.001$.

Footnotes

1. Officially given in Exercise 36. ↩