Chapter9: Sequences, Probability And Counting Theory
Section9.4: Series And Their Notations
Problem 55SE: The sum of an infinite geometric series is five times the value of the first term. What is the...
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Question
Please do the following for this series;
A) Write this series in “expanded form”
(B) Write out the first few terms of the sequence of partial sums.
![This image represents an infinite series in mathematics. The series is expressed as:
\[ \sum_{n=1}^{\infty} \frac{1}{2^n + 1} \]
Explanation:
- The symbol \(\sum\) denotes summation.
- The index of summation is \(n\), which starts at 1 and goes to infinity, as indicated by the limits below and above the summation symbol, respectively.
- The summand (term to sum) is \(\frac{1}{2^n + 1}\), meaning for each integer \(n\) starting from 1 and increasing indefinitely, you calculate \(2^n + 1\) and then take the reciprocal of that result.
This series adds the reciprocal of \(2^n + 1\) for each \(n\) from 1 to infinity.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5f5d57bb-887b-451c-add8-753065e065d0%2Fb8ce8e83-184f-4b28-9db1-a32a6e2efd9d%2Fetey36_processed.png&w=3840&q=75)
Transcribed Image Text:This image represents an infinite series in mathematics. The series is expressed as:
\[ \sum_{n=1}^{\infty} \frac{1}{2^n + 1} \]
Explanation:
- The symbol \(\sum\) denotes summation.
- The index of summation is \(n\), which starts at 1 and goes to infinity, as indicated by the limits below and above the summation symbol, respectively.
- The summand (term to sum) is \(\frac{1}{2^n + 1}\), meaning for each integer \(n\) starting from 1 and increasing indefinitely, you calculate \(2^n + 1\) and then take the reciprocal of that result.
This series adds the reciprocal of \(2^n + 1\) for each \(n\) from 1 to infinity.
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