/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 1 Find a formula for the \(n\) th ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

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, and the sum is 3.

Step by step solution

01

Identify the type of series

The given series is \(2 + \frac{2}{3} + \frac{2}{9} + \frac{2}{27} + \dots\). We can observe that each term is formed by multiplying the previous term by \(\frac{1}{3}\). Therefore, this is a geometric series with the first term \(a = 2\) and a common ratio \(r = \frac{1}{3}\).
02

Write the formula for the partial sum of a geometric series

For a geometric series with first term \(a\) and common ratio \(r\), the formula for the sum of the first \(n\) terms is given by:\[S_n = a \frac{1 - r^n}{1 - r}\]Substituting the given values: \(a = 2\) and \(r = \frac{1}{3}\), the formula becomes:\[S_n = 2 \frac{1 - (\frac{1}{3})^n}{1 - \frac{1}{3}}\]
03

Simplify the formula for the partial sum

Simplify the denominator: \(1 - \frac{1}{3} = \frac{2}{3}\). Thus, the partial sum becomes:\[S_n = 2 \times \frac{1 - (\frac{1}{3})^n}{\frac{2}{3}} = 3 \left(1 - \left(\frac{1}{3}\right)^n\right)\]
04

Determine the series' sum as \(n\) approaches infinity

A geometric series converges when the absolute value of the common ratio \(|r| < 1\). For this series, \(r = \frac{1}{3}\), which is less than 1, so it converges. The sum to infinity is:\[S = \frac{a}{1 - r} = \frac{2}{1 - \frac{1}{3}} = \frac{2}{\frac{2}{3}} = 3\]
05

Conclusion

We found the formula for the \(n\)th partial sum to be \(S_n = 3 \left(1 - \left(\frac{1}{3}\right)^n\right)\), and the sum of the series as \(n\) approaches infinity is 3.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Partial Sum
In mathematics, especially when dealing with sequences and series, a partial sum is an important concept. A partial sum is simply the sum of the first few terms of a series, up to a specified point. For example, if you have the series:
  • 2,
  • \(\frac{2}{3}\),
  • \(\frac{2}{9}\),
  • \(\frac{2}{27}\),\(...\),
the partial sum involves adding these terms up until a certain number of terms, say the first five terms. The partial sum is useful for understanding how the series builds up.
For geometric series, the formula for the \(n\)th partial sum is given by:\[S_n = a \frac{1 - r^n}{1 - r}\]where \(a\) is the first term and \(r\) is the common ratio. Substituting specific values helps find the sum up to the \(n\)th term.
Series Convergence
Convergence is a critical concept in the study of series. When a series converges, the sum of its infinite terms approaches a specific number, rather than becoming infinitely large or oscillating without bound.
In the case of geometric series, convergence is determined by the common ratio \(r\). Specifically, a geometric series converges if the absolute value of \(r\) is less than 1, i.e., \(|r| < 1\).
For the series we examined
  • The first term \(a = 2\).
  • The common ratio \(r = \frac{1}{3}\),
we can see that it converges because \(\frac{1}{3} < 1\). When a geometric series converges, its sum to infinity can be computed using the formula:\[S = \frac{a}{1 - r}\]This allows us to calculate the exact sum to which the series converges.
Common Ratio
The common ratio is a fundamental element of a geometric series. It is the constant factor between consecutive terms in a geometric sequence or series. You can find it by dividing any term by its preceding term.
For example, with the series 2, \(\frac{2}{3}\), \(\frac{2}{9}\), \(\frac{2}{27}\), and so on, each term multiplied by \(\frac{1}{3}\) gives the next.
  • If the first term is 2, then \(2 \times \frac{1}{3} = \frac{2}{3}\).
  • Continues with \(\frac{2}{3} \times \frac{1}{3} = \frac{2}{9}\), and so forth.
This constant \(\frac{1}{3}\) is the common ratio \(r\). Knowing the common ratio is crucial for deriving and calculating the partial sums, as well as for determining the series' convergence.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the series' radius of convergence. $$\sum_{n=1}^{\infty}\left(\frac{2 \cdot 4 \cdot 6 \cdots(2 n)}{2 \cdot 5 \cdot 8 \cdot \cdots(3 n-1)}\right)^{2} x^{n}$$

Which of the series \(\Sigma_{n=1}^{\infty} a_{n}\) defined by the formulas converge, and which diverge? Give reasons for your answers. $$a_{1}=\frac{1}{3}, \quad a_{n+1}=\sqrt[n]{a_{n}}$$

Use any method to determine whether the series converges or diverges. Give reasons for your answer. $$\sum_{n=1}^{\infty} \frac{(2 n+3)\left(2^{n}+3\right)}{3^{n}+2}$$

Which of the series \(\Sigma_{n=1}^{\infty} a_{n}\) defined by the formulas converge, and which diverge? Give reasons for your answers. $$a_{1}=1, \quad a_{n+1}=\frac{1+\ln n}{n} a_{n}$$

Assume that \(b_{n}\) is a sequence of positive numbers converging to 4/5. Determine whether the following series converge or diverge. a. \(\sum_{n=1}^{\infty}\left(b_{n}\right)^{1 / n}\) b. \(\sum_{n=1}^{\infty}\left(\frac{5}{4}\right)^{n}\left(b_{n}\right)\) c. \(\sum_{n=1}^{\infty}\left(b_{n}\right)^{n}\) d. \(\sum_{n=1}^{\infty} \frac{1000^{n}}{n !+b_{n}}\)
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.