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Chapter 9: Problem 1

Find a formula for the \(n\) th partial sum of each series and use it to find the series' sum if the series converges. $$2+\frac{2}{3}+\frac{2}{9}+\frac{2}{27}+\dots+\frac{2}{3^{n-1}}+\dots$$

Short Answer

Expert verified
The series converges to a sum of 3.

Step by step solution

01

Identify the Series Type and First Term

The series is given as \(2+\frac{2}{3}+\frac{2}{9}+\frac{2}{27}+\dots\). Recognize this as a geometric series with the first term \(a = 2\).
02

Find the Common Ratio

The common ratio \(r\) of the series is the factor by which each term is multiplied to get the next term. From \(2\) to \(\frac{2}{3}\), we divide by 3, which gives a common ratio \(r = \frac{1}{3}\).
03

Formula for the n-th Partial Sum of a Geometric Series

The formula for the n-th partial sum \(S_n\) of a geometric series is given by: \[ S_n = a \frac{1-r^n}{1-r} \] where \(a = 2\) and \(r = \frac{1}{3}\).
04

Substitute Values into the Partial Sum Formula

Substitute \(a = 2\) and \(r = \frac{1}{3}\) into the partial sum formula: \[ S_n = 2 \frac{1-\left(\frac{1}{3}\right)^n}{1-\frac{1}{3}} = 2 \frac{1-\left(\frac{1}{3}\right)^n}{\frac{2}{3}} \].
05

Simplify the Expression

Simplify the expression: \[ S_n = 2 \cdot \frac{3}{2} \left(1-\left(\frac{1}{3}\right)^n\right) = 3 \left(1-\left(\frac{1}{3}\right)^n\right) \].
06

Determine Convergence

As \(n\) approaches infinity, \(\left(\frac{1}{3}\right)^n\) approaches 0. Therefore, the series converges to \(S = 3 \left(1-0\right) = 3\).

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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 special kind of sequence in mathematics. Each term in the sequence is obtained by multiplying the previous term by a constant, known as the common ratio.
For example, in the series given:
  • the first term is 2
  • the common ratio is \( \frac{1}{3} \)
This means every term is \( \frac{1}{3} \) times the term before it. Starting from 2, it becomes \( \frac{2}{3} \), then \( \frac{2}{9} \), continuing indefinitely.
When dealing with geometric series, identifying the first term \(a\) and the common ratio \(r\) is crucial as they help us understand the characteristics of the series and find its sums or other properties.
Partial Sum
Understanding the partial sum of a geometric series is essential, especially when you want to calculate the sum of a specific number of terms.
The partial sum \(S_n\) of the first \(n\) terms of a geometric series is obtained with a specific formula:
  • \( S_n = a \frac{1-r^n}{1-r} \)
For the series at hand, substituting the first term \(a = 2\) and the common ratio \(r = \frac{1}{3}\) into this formula allows us to find the sum of any specified number of terms: \[ S_n = 3 \left(1-\left(\frac{1}{3}\right)^n\right) \]This compact expression tells us the sum of the first \(n\) terms of the sequence, making it easier to study the series incrementally. As the partial sum can be evaluated for any \(n\), it provides a step-by-step way to see how the series grows.
Series Convergence
Series convergence refers to the behavior of a series as it progresses towards infinity. Specifically, it observes whether the series approaches a finite limit or diverges.
In the context of geometric series, the series converges if the absolute value of the common ratio \(r\) is less than 1.
This rule ensures the series heads to a finite limit rather than growing indefinitely. For the presented series, with \(r = \frac{1}{3}\),
  • \( |r| = \frac{1}{3} < 1 \),
indicating convergence.Moreover, as \(n\) increases, the term \( \left(\frac{1}{3}\right)^n \) approaches 0, simplifying the sum of the series. Consequently, the convergent sum of the series can be determined as 3 when \(n\) tends to infinity.
Understanding this concept is vital, as it tells us that despite having an infinite number of terms, we can still compute a closed sum.

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Most popular questions from this chapter

$$\text { Let } a_{n}=\left\\{\begin{array}{ll} n / 2^{n}, & \text { if } n \text { is a prime number } \\ 1 / 2^{n}, & \text { otherwise. } \end{array}\right.$$ Does \(\Sigma a_{n}\) converge? Give reasons for your answer.

Prove that if \(\left\\{a_{n}\right\\}\) is a convergent sequence, then to every positive number \(\epsilon\) there corresponds an integer \(N\) such that for all \(m\) and \(n\), $$m>N \quad \text { and } \quad n>N \Rightarrow \quad\left|a_{m}-a_{n}\right|<\epsilon.$$

Find the values of \(x\) for which the given geometric series converges. Also, find the sum of the series (as a function of \(x\) ) for those values of \(x .\) \( \)\sum_{n=0}^{\infty}(\ln x)^{n}$$

If \(\Sigma a_{n}\) converges and \(a_{n}>0\) for all \(n,\) can anything be said about \(\Sigma\left(1 / a_{n}\right) ?\) Give reasons for your answer.

Taylor's formula with \(n=1\) and \(a=0\) gives the linearization of a function at \(x=0 .\) With \(n=2\) and \(n=3\) we obtain the standard quadratic and cubic approximations. In these exercises we explore the errors associated with these approximations. We seek answers to two questions: a. For what values of \(x\) can the function be replaced by each approximation with an error less than \(10^{-2} ?\). b. What is the maximum error we could expect if we replace the function by each approximation over the specified interval? Using a CAS, perform the following steps to aid in answering questions (a) and (b) for the functions and intervals. Step \(I:\) Plot the function over the specified interval. Step 2: Find the Taylor polynomials \(P_{1}(x), P_{2}(x),\) and \(P_{3}(x)\) at \(x=0\) Step 3: Calculate the \((n+1)\) st derivative \(f^{(n+1)}(c)\) associated with the remainder term for each Taylor polynomial. Plot the derivative as a function of \(c\) over the specified interval and estimate its maximum absolute value, \(M .\) Step 4: Calculate the remainder \(R_{n}(x)\) for each polynomial. Using the estimate \(M\) from Step 3 in place of \(f^{(n+1)}(c),\) plot \(R_{n}(x)\) over the specified interval. Then estimate the values of \(x\) that answer question (a). Step 5: Compare your estimated error with the actual error \(E_{n}(x)=\left|f(x)-P_{n}(x)\right|\) by plotting \(E_{n}(x)\) over the specified interval. This will help answer question (b). Step 6: Graph the function and its three Taylor approximations together. Discuss the graphs in relation to the information discovered in Steps 4 and 5. $$f(x)=e^{-x} \cos 2 x, \quad|x| \leq 1$$
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