### Home > APCALC > Chapter 9 > Lesson 9.1.3 > Problem9-44

9-44.

Determine whether each of the following infinite series converges or diverges. Explain each answer briefly. If the series converges, calculate its exact sum.

1. $\displaystyle \sum _ { k = 1 } ^ { \infty } 1$

What is $1 + 1 + 1 + 1 + ...$ ?

1. $\displaystyle\sum _ { j = 2 } ^ { \infty } 8 ( \frac { 3 } { 4 } ) ^ { j }$

This is a geometric sequence with a ratio of $3/4$.

1. $\displaystyle\sum _ { k = 0 } ^ { \infty } 20 ( 0.4 ) ^ { k }$

This is also a geometric sequence. What is the ratio? (See part (b).)

1. $\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 3 } { n ( n + 1 ) }$

This can be written as a telescoping sum:

$S=3\bigg(\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+...\bigg)$

$\text{ }=3(1-\frac{1}{2}+(\frac{1}{2}-\frac{1}{3})+(\frac{1}{3}-\frac{1}{4})+...)$