### Home > APCALC > Chapter 9 > Lesson 9.1.1 > Problem9-15

9-15.

Ying has a different method for summing an infinite series. Here are her steps:

• 1) Let $S =$ the sum.

• 2) Separate the first term from the rest of the series.

• 3) Factor the rest of the terms so that the remaining factor is $S$.

• 4) Substitute $S$ for the other factor.

• 5) Solve for $S$.

Apply Ying’s method to the following problem. The first three steps are done for you. Copy them, then complete the solution.

 Problem:Start of Solution: $\left. \begin{array}{l}{\ \ \ \ \ \ \ \ \ \ \ \ \frac { 1 } { 2 } + \frac { 1 } { 6 } + \frac { 1 } { 18 } + \ldots }\\\text{1) }{ S = \frac { 1 } { 2 } + \frac { 1 } { 6 } + \frac { 1 } { 18 } + \ldots }\\\text{2) }{ S = \frac { 1 } { 2 } + ( \frac { 1 } { 6 } + \frac { 1 } { 18 } + \ldots ) }\\\text{3) }{ S = \frac { 1 } { 2 } + \frac { 1 } { 3 } ( \frac { 1 } { 2 } + \frac { 1 } { 6 } + \frac { 1 } { 18 } + \ldots ) }\end{array} \right.$

$S=\frac{1}{2}+\frac{1}{3}S$