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Chapter 13: Problem 11

Find the sum of each series. $$\sum_{i=1}^{5} 2^{-i}$$

Short Answer

Expert verified
\(\frac{31}{32}\)

Step by step solution

01

Identify the series

The series in question is \(\sum_{i=1}^{5} 2^{-i}\). This represents the sum of the terms when the exponent runs from 1 to 5, with each term being \(2^{-i}\).
02

Write out the first few terms

List the first few terms of the series: \(2^{-1}, 2^{-2}, 2^{-3}, 2^{-4}, 2^{-5}\). This translates to \(\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32}\).
03

Sum the terms manually

Add each of the terms together step by step: \(\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32}\). To simplify this, it helps to use a common denominator: 32. Convert each fraction: \(\frac{16}{32} + \frac{8}{32} + \frac{4}{32} + \frac{2}{32} + \frac{1}{32}\). Adding these together: \(16 + 8 + 4 + 2 + 1 = 31\), thus the sum is \(\frac{31}{32}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

geometric series
A geometric series is one where each term is obtained by multiplying the previous term by a constant factor known as the common ratio. In our problem, the series is \(\sum_{i=1}^{5} 2^{-i}\). Each term here decreases by a factor of \(2\).
For instance:
  • 1st term: \(2^{-1} = \frac{1}{2}\)
  • 2nd term: \(2^{-2} = \frac{1}{4}\)
  • 3rd term: \(2^{-3} = \frac{1}{8}\)
  • 4th term: \(2^{-4} = \frac{1}{16}\)
  • 5th term: \(2^{-5} = \frac{1}{32}\)
Understanding geometric series helps us to comprehend how a sequence of numbers progresses and the nature of their sums.
fraction addition
Adding fractions involves finding a common denominator. This allows us to sum the numerators directly while keeping the denominator the same.
For instance, to add \(\frac{1}{2}\) and \(\frac{1}{4}\), convert each fraction to have a common denominator of 4:
\(\frac{1}{2} = \frac{2}{4}\) Next, add the numerators: \(\frac{2}{4} + \frac{1}{4} = \frac{3}{4}\)
In our problem, summing all terms required converting each fraction to a denominator of 32:
  • \(\frac{1}{2} = \frac{16}{32}\)
  • \(\frac{1}{4} = \frac{8}{32}\)
  • \(\frac{1}{8} = \frac{4}{32}\)
  • \(\frac{1}{16} = \frac{2}{32}\)
  • \(\frac{1}{32} = \frac{1}{32}\)
Adding these fractions gives us \(\frac{31}{32}\) as the sum.
partial sums
A partial sum is simply the sum of the first 'n' terms of a series. In our case, it's the sum of the first 5 terms:
  • \(\frac{1}{2}\)
  • \(\frac{1}{4}\)
  • \(\frac{1}{8}\)
  • \(\frac{1}{16}\)
  • \(\frac{1}{32}\)
The total is computed by adding each fraction together, ensuring they are converted to a common denominator for ease:
  • \(\frac{16}{32} + \frac{8}{32} + \frac{4}{32} + \frac{2}{32} + \frac{1}{32} = \frac{31}{32}\)
The concept of partial sums is useful as it helps in understanding the behavior of a series as more terms are added.

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Most popular questions from this chapter

List all terms of each finite sequence. \(c_{n}=(-3)^{n-2}\) for \(1 \leq n \leq 5\)

Use a series to model the situation in each of the following problems. A frog with a vision problem is 1 yard away from a dead cricket. He spots the cricket and jumps halfway to the cricket. After the frog realizes that he has not reached the cricket, he again jumps halfway to the cricket. Write a series in summation notation to describe how far the frog has moved after nine such jumps.

Write a formula for the general term of each infinite sequence. \(4,8,12,16, \dots\)

Working in groups, have someone in each group make up a formula for \(a_{n},\) the \(n\)th term of a sequence, but do not show it to the other group members. Write the terms of the sequence on a piece of paper one at a time. After each term is given, ask whether anyone knows the next term. When the group can correctly give the next term, ask for a formula for the \(n\)th term.

What is the difference between a sequence and a series?
See all solutions

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