text: 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 $\displaystyle a=\tfrac{1}{2}$ then $\displaystyle 1+a+...+a^n = \frac{1-a^n}{1-a} = \frac{1}{1-a} - \frac{a^n}{1-a}$
  
  Therefore, $\displaystyle \left| \sum_{k=0}^n a^k - 2 \right| \leq \left(\frac{1}{2}\right)^{n-1}$
  
  Now make the error term sufficiently small.