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Chapter 9: Problem 46

Find the sum of the convergent series. \(4-2+1-\frac{1}{2}+\cdots\)

Short Answer

Expert verified
The sum of the infinite geometric series \(4-2+1-\frac{1}{2}+\cdots\) is 8/3.

Step by step solution

01

Identify the first term, common ratio, and assess convergence

In the series \(4-2+1-\frac{1}{2}+\cdots\), the first term 'a' is 4 and the common ratio 'r' is -1/2. To check the convergence of the series, check if the absolute value of 'r' is less than 1. Here, we have |-1/2| = 0.5 which is less than 1. Therefore, the given series converges.
02

Apply the sum formula for a convergent geometric series

The sum 'S' of an infinite convergent geometric series is given by the formula \(S=a/(1-r)\). Substituting a = 4 and r = -1/2 into this formula, we get \(S = 4 / (1 - (-1/2))\).
03

Simplify the resulting expression

After simplifying the expression obtained in step 2, it reduces to \(S = 4 / (1.5)\) which simplifies further to yield \(S = 8/3\).

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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 a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. In the series given in the exercise,
  • the first term is 4,
  • the second term is -2,
  • the third term is 1,
  • and so on.
Each term progressively decreases in size, oscillating between positive and negative. Geometric series can either be finite, with a specific number of terms, or infinite, continuing indefinitely. The series in the exercise is infinite because it continues with a pattern without end.
Common Ratio
The common ratio in a geometric series is the factor by which we multiply each term to get the next term. It remains constant throughout the series. Identifying the common ratio is crucial because it determines the behavior of the series. For the series in the exercise:
  • The first term (4) is multiplied by -1/2 to get the second term (-2).
  • Continuing this pattern shows the common ratio 'r' is -1/2.
In any series, this ratio can be found by dividing any term by its preceding term. For this series, \[r = \frac{-2}{4} = -\frac{1}{2}\]Given that our series oscillates, the common ratio of -1/2 is negative, indicating alternating signs between terms. The absolute value of the common ratio is less than 1, which is necessary for the series to be classified as convergent.
Convergence
Convergence in the context of series refers to whether the series approaches a specific value as the number of terms increases. For geometric series, the series converges if the absolute value of the common ratio is less than 1. When this condition is met, the series will approach a final sum as more terms are added, instead of diverging to infinity.In the exercise,
  • we confirmed convergence by checking the common ratio鈥檚 absolute value, |-1/2| = 0.5.
  • Since 0.5 is less than 1, the series converges.
Once we know a series converges, we can use the formula for the sum of an infinite geometric series, \[S = \frac{a}{1 - r}\]where \(a\) is the first term and \(r\) is the common ratio. Using this formula ensures we find the exact sum the series approaches after an infinite number of terms, as demonstrated with the answer \(\frac{8}{3}\) in this exercise.

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Most popular questions from this chapter

Write a power series that has the indicated interval of convergence. Explain your reasoning. (a) \((-2,2)\) (b) \((-1,1]\) (c) \((-1,0)\) (d) \([-2,6)\)

Write \(\sum_{k=1}^{\infty} \frac{6^{k}}{\left(3^{k+1}-2^{k+1}\right)\left(3^{k}-2^{k}\right)}\) as a rational number.

Probability In an experiment, three people toss a fair coin one at a time until one of them tosses a head. Determine, for each person, the probability that he or she tosses the first head. Verify that the sum of the three probabilities is 1 .

Consider the sequence \(\left\\{a_{n}\right\\}=\left\\{n r^{n}\right\\}\). Decide whether \(\left\\{a_{n}\right\\}\) converges for each value of \(r\). (a) \(r=\frac{1}{2}\) (b) \(r=1\) (c) \(r=\frac{3}{2}\) (d) For what values or \(r\) does the sequence \(\left\\{n r^{n}\right\\}\) converge?

The series represents a well-known function. Use a computer algebra system to graph the partial sum \(S_{10}\) and identify the function from the graph. $$ f(x)=\sum_{n=0}^{\infty}(-1)^{n} \frac{x^{2 n}}{(2 n) !} $$
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