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Chapter 10: Problem 7

Write out the first eight terms of each series to show how the series starts. Then find the sum of the series or show that it diverges. $$\sum_{n=0}^{\infty} \frac{(-1)^{n}}{4^{n}}$$

Short Answer

Expert verified
The series converges to \(\frac{4}{5}\).

Step by step solution

01

Identify the Series

The given series is \(\sum_{n=0}^{\infty} \frac{(-1)^{n}}{4^{n}}\). This is an infinite series with each term given by the formula \(a_n = \frac{(-1)^n}{4^n}\).
02

Write the First Eight Terms

To find the first eight terms, substitute \(n = 0, 1, 2, \ldots, 7\) into the formula \(a_n = \frac{(-1)^n}{4^n}\):1. \(n=0\): \(\frac{(-1)^0}{4^0} = 1\)2. \(n=1\): \(\frac{(-1)^1}{4^1} = -\frac{1}{4}\)3. \(n=2\): \(\frac{(-1)^2}{4^2} = \frac{1}{16}\)4. \(n=3\): \(\frac{(-1)^3}{4^3} = -\frac{1}{64}\)5. \(n=4\): \(\frac{(-1)^4}{4^4} = \frac{1}{256}\)6. \(n=5\): \(\frac{(-1)^5}{4^5} = -\frac{1}{1024}\)7. \(n=6\): \(\frac{(-1)^6}{4^6} = \frac{1}{4096}\)8. \(n=7\): \(\frac{(-1)^7}{4^7} = -\frac{1}{16384}\)The first eight terms are: \(1, -\frac{1}{4}, \frac{1}{16}, -\frac{1}{64}, \frac{1}{256}, -\frac{1}{1024}, \frac{1}{4096}, -\frac{1}{16384}\).
03

Identify the Type of Series

The series is a geometric series with the first term \(a = 1\) and common ratio \(r = -\frac{1}{4}\).
04

Determine if the Series Converges

A geometric series converges if the absolute value of the common ratio \(|r|\) is less than 1. Here, \(|r| = \left| -\frac{1}{4} \right| = \frac{1}{4} < 1\), so the series converges.
05

Calculate the Sum of the Series

For a convergent geometric series, the sum \(S\) can be calculated using the formula:\[ S = \frac{a}{1-r} \]Substituting \(a = 1\) and \(r = -\frac{1}{4}\) gives:\[ S = \frac{1}{1 - (-\frac{1}{4})} = \frac{1}{1 + \frac{1}{4}} = \frac{1}{\frac{5}{4}} = \frac{4}{5} \]
06

Conclusion: Present the Sum

The sum of the series \(\sum_{n=0}^{\infty} \frac{(-1)^{n}}{4^{n}}\) is \(\frac{4}{5}\).

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Key Concepts

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

Series Convergence
A series is a convergence when its terms approach a specific value as the number of terms increases. For geometric series, we have a simple criterion. If the absolute value of the common ratio \(|r|\) is less than 1, the series converges.
This means the sum will eventually settle at a certain value rather than increasing indefinitely.
In our example series with formula \(a_n = \frac{(-1)^n}{4^n}\), the common ratio \(r\) is \(-\frac{1}{4}\). Since \(|-\frac{1}{4}| = \frac{1}{4} < 1\), we can conclude that this geometric series converges.
  • Convergent series result in finite sums.
  • Understanding convergence helps in predicting long-term behavior of series.
Infinite Series
An infinite series is a sum of an infinite sequence of numbers. Each term in the series comes from a pattern defined by a formula.
The infinite series can be written in the form \( \sum_{n=0}^{\infty} a_n \), where \(a_n\) represents each term of the series.
In our exercise, the given series is \( \sum_{n=0}^{\infty} \frac{(-1)^n}{4^n} \). Each term alternates in sign and becomes smaller as \(n\) increases, indicating a geometric nature.
  • Infinite series can be convergent or divergent.
  • For geometric series, identifying the convergence determines if a sum exists.
Sum of Series
The sum of a series is the value approached by this sum as more terms are added. In the case of convergent geometric series, the sum can be calculated using a specific formula.
For a geometric series starting with term \(a\) and common ratio \(r\), the sum \(S\) is given by:
\[ S = \frac{a}{1-r} \] For the provided series \( \sum_{n=0}^{\infty} \frac{(-1)^n}{4^n} \), we have a first term \(a = 1\) and a ratio \(r = -\frac{1}{4}\).
To find the sum, substitute these values into the formula:\[ S = \frac{1}{1 - (-\frac{1}{4})} = \frac{1}{1 + \frac{1}{4}} = \frac{1}{\frac{5}{4}} = \frac{4}{5} \]
  • This result shows the sum approaches \(\frac{4}{5}\), indicating the series converges to this value.
  • Calculating sums helps provide the expected total value of a convergent series.

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Most popular questions from this chapter

Assume that each sequence converges and find its limit. \(a_{1}=-1, \quad a_{n+1}=\frac{a_{n}+6}{a_{n}+2}\)

Determine if the sequence is monotonic and if it is bounded. \(a_{n}=\frac{3 n+1}{n+1}\)

Use the definition of convergence to prove the given limit. \(\lim _{n \rightarrow \infty}\left(1-\frac{1}{n^{2}}\right)=1\)

a. Series for \(\sinh ^{-1} x \quad\) Find the first four nonzero terms of the Taylor series for $$ \sinh ^{-1} x=\int_{0}^{x} \frac{d t}{\sqrt{1+t^{2}}} $$ b. Use the first three terms of the series in part (a) to estimate \(\sinh ^{-1} 0.25 .\) Give an upper bound for the magnitude of the estimation error.

If the terms of one sequence appear in another sequence in their given order, we call the first sequence a sub-sequence of the second. Prove that if two sub-sequences of a sequence \(\left\\{a_{n}\right\\}\) have different limits \(L_{1} \neq L_{2}\) then \(\left\\{a_{n}\right\\}\) diverges.
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