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Chapter 9: Problem 85

Find all values of \(x\) for which the series converges. For these values of \(x\), write the sum of the series as a function of \(x\). $$ \sum_{n=0}^{\infty}\left(\frac{1}{x}\right)^{n} $$

Short Answer

Expert verified
The series \(\sum_{n=0}^{\infty}\left(\frac{1}{x}\right)^{n}\) converges for \(x > 1\) and \(x < -1\). When it does converge, the sum of the series is \(\frac{1}{1 - \frac{1}{x}}\).

Step by step solution

01

Define the series and the ratio

We are working with the series \(\sum_{n=0}^{\infty}\left(\frac{1}{x}\right)^{n}\), and the common ratio is \(\frac{1}{x}\).
02

Apply the Geometric Series Test

The Geometric Series Test tells us a geometric series converges if \(|\frac{1}{x}| < 1\). We solve this inequality to find the values of \(x\) for which the series converges.
03

Find the sum for the converging series

For a geometric series, the sum is given by \(\frac{a}{1 - r}\), where \(a\) is the first term and \(r\) is the ratio. Since the first term of our series is \(\left(\frac{1}{x}\right)^0 = 1\), the sum of the series, when it converges, is \(\frac{1}{1 - \frac{1}{x}}\).

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Key Concepts

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

Series Convergence
Understanding whether a series converges is crucial in mathematics, especially when dealing with infinite series. A series converges if the sum of its terms approaches a specific value as more terms are added. In contrast, if the sum continues to grow indefinitely or oscillates, the series does not converge.

For the given series, \[ \sum_{n=0}^{\infty}\left(\frac{1}{x}\right)^{n}, \] the convergence depends on the behavior of the common ratio as we progress through the series. If the ratio's absolute value is less than 1, the series' terms get smaller and smaller, leading the series to converge.
Geometric Series Test
The Geometric Series Test helps determine if an infinite geometric series converges. A geometric series has a constant ratio between successive terms. To use this test, we check if the absolute value of this ratio is less than 1.

For example, in the series \[ \sum_{n=0}^{\infty}\left(\frac{1}{x}\right)^{n}, \] the common ratio is \( \frac{1}{x} \). Thus, the series converges when \( \left|\frac{1}{x}\right| < 1 \).

Solving this inequality will help find the range of \( x \) values for which the series converges.
Inequality Solving
To determine the values of \( x \) for which the series converges, we solve the inequality \(|\frac{1}{x}| < 1\).

This inequality can be broken down as follows:
  • \( -1 < \frac{1}{x} < 1 \)
Solving these inequalities separately gives:
  • \( \frac{1}{x} > -1 \) implies \( x > -1 \)
  • \( \frac{1}{x} < 1 \) implies \( x > 1 \)
Combining these results, \( x \) must be greater than 1 for the series to converge.
Series Sum Formula
Once we establish that the series converges, we can compute its sum using the series sum formula, \[ S = \frac{a}{1 - r}, \] where \( a \) is the first term, and \( r \) is the common ratio.

In the context of \[ \sum_{n=0}^{\infty}\left(\frac{1}{x}\right)^{n}, \] the first term \( a \) is 1 because \( \left(\frac{1}{x}\right)^{0} = 1 \). The ratio \( r \) is \( \frac{1}{x} \).

Hence, the sum of the series, given it converges, is \[ S = \frac{1}{1 - \frac{1}{x}}, \] valid for \( x > 1 \). This formula is beneficial for finding the sum of terms in converging geometric series.

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Most popular questions from this chapter

(a) Show that \(\int_{1}^{n} \ln x d x<\ln (n !)\) for \(n \geq 2\). (b) Draw a graph similar to the one above that shows \(\ln (n !)<\int_{1}^{n+1} \ln x d x\) (c) Use the results of parts (a) and (b) to show that \(\frac{n^{n}}{e^{n-1}}1 .\) (d) Use the Squeeze Theorem for Sequences and the result of part (c) to show that \(\lim _{n \rightarrow \infty} \frac{\sqrt[n]{n !}}{n}=\frac{1}{e^{*}}\) (e) Test the result of part (d) for \(n=20,50\), and 100 .

Depreciation A company buys a machine for \(\$ 225,000\) that depreciates at a rate of \(30 \%\) per year. Find a formula for the value of the machine after \(n\) years. What is its value after 5 years?

Use the formula for the \(n\) th partial sum of a geometric series \(\sum_{i=0}^{n-1} a r^{i}=\frac{a\left(1-r^{n}\right)}{1-r}\) Present Value The winner of a \(\$ 1,000,000\) sweepstakes will be paid \(\$ 50,000\) per year for 20 years. The money earns \(6 \%\) interest per year. The present value of the winnings is \(\sum_{n=1}^{20} 50,000\left(\frac{1}{1.06}\right)^{n} .\) Compute the present value and interpret its meaning.

True or False? Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. Consider the sequence \(\sqrt{6}, \sqrt{6+\sqrt{6}}, \sqrt{6+\sqrt{6+\sqrt{6}}}, \ldots\) (a) Compute the first five terms of this sequence. (b) Write a recursion formula for \(a_{n}\), for \(n \geq 2\). (c) Find lim \(a_{n}\).

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. $$ f(x)=\sum_{n=0}^{\infty}(-1)^{n} \frac{x^{2 n+1}}{(2 n+1) !} $$
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