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.
explanation: The document contains a math question and an answer to that question. The question asks how many terms of the geometric series \(1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots\) are needed to be within \( \epsilon = 0.0001 \) of the target value 2. The answer provides a detailed mathematical derivation to solve this problem.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: 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.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question and an answer to that question. The question asks how many terms of the geometric series \(1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots\) are needed to be within \( \epsilon = 0.0001 \) of the target value 2. The answer provides a detailed mathematical derivation to solve this problem.
  
  Label: question-and-answer