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.
The text contains a mathematical question about the number of terms needed in a geometric series to be within a certain error margin of a target value, and it provides a detailed solution to the problem.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
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.
user
The text contains a mathematical question about the number of terms needed in a geometric series to be within a certain error margin of a target value, and it provides a detailed solution to the problem.

TYPE: SINGLE_QUESTION_WITH_ANSWER