### Home > APCALC > Chapter 9 > Lesson 9.4.2 > Problem9-127

9-127.

Timmy is tired. He does not want to add the infinite terms of $\frac { 1 } { 2 } + \frac { 1 } { 4 } + \frac { 1 } { 8 } + \frac { 1 } { 16 } + \ldots$. So instead, he just adds the first three terms.

1. How far is his result from the actual sum of the infinite series? (This is called his “error.”)

For an infinite geometric series:

$S=\frac{a}{1-r}$

2. What would his error have been if he added the first four terms?

3. Generalize his error. That is, if he adds up $n$ terms of this geometric series, what will his error be?

For a finite geometric series:

$S_n=\frac{a(1-r^n)}{1-r}$

$\text{error}=|S-S_n|$

Substitute $a = 1/2$ and $r = 1/2$ into the formulas in the hints then evaluate.