1. analysis 4.1.1

4.1.1

How many terms of the series
1 + 1/2 + 1/4 + 1/8 + ...
do we need to take if we are to guarantee that we are within є = 0.0001 of the target value 2?

2. Originally Posted by pila0688
How many terms of the series
1 + 1/2 + 1/4 + 1/8 + ...
do we need to take if we are to guarantee that we are within є = 0.0001 of the target value 2?
Let $a=\tfrac{1}{2}$ then $1+a+...+a^n = \frac{1-a^n}{1-a} = \frac{1}{1-a} - \frac{a^n}{1-a}$

Therefore, $\left| \sum_{k=0}^n a^k - 2 \right| \leq
\left(\frac{1}{2}\right)^{n-1}$

Now make the error term sufficiently small.