CPM Homework Help : CALC Problem 9-45

Determine whether each of the infinite series converges or diverges. Explain each answer briefly. If the series converges, find its exact sum. Homework Help ✎

Hint:

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

Hint:

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

Hint:

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

Steps:

This can be written as a telescoping sum:

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

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