/*! 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 26 Evaluate the geometric series. ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 26

Evaluate the geometric series. $$ 1-\frac{1}{3}+\frac{1}{9}-\frac{1}{27}+\cdots+\frac{1}{3^{60}}-\frac{1}{3^{61}}+\frac{1}{3^{62}} $$

Short Answer

Expert verified
The sum of the given geometric series is \(S_{63} = \frac{3(1-(-\frac{1}{3})^{63})}{4}\).

Step by step solution

01

Identify the first term, the common ratio, and the number of terms.

From the given series, we can see that the first term is \(a_1 = 1\), the common ratio is \(r = -\frac{1}{3}\), and since the series goes up to \(\frac{1}{3^{62}}\), we can infer that the number of terms is 63.
02

Apply the formula for the sum of the finite geometric series.

Now that we have identified the first term, the common ratio, and the number of terms, we can use the formula for the sum of a finite geometric series: $$ S_n = \frac{a_1(1-r^n)}{1-r}. $$ Plug the values we found in step 1 into the formula: $$ S_{63} = \frac{1(1-(-\frac{1}{3})^{63})}{1-(-\frac{1}{3})}. $$
03

Simplify the expression.

Simplify the expression as follows: $$ S_{63} = \frac{1(1-(-\frac{1}{3})^{63})}{\frac{4}{3}} = \frac{3(1-(-\frac{1}{3})^{63})}{4}. $$
04

Evaluate the geometric series.

Now we can evaluate the sum of the given geometric series: $$ S_{63} = \frac{3(1-(-\frac{1}{3})^{63})}{4}. $$ This is the final expression for the sum of the given geometric series.

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!

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

Write the series explicitly and evaluate the sum. $$ \sum_{m=1}^{5}\left(m^{2}-2 m+7\right) $$

In the decimal expansion of \(0.9^{9999}\), how many zeros follow the decimal point before the first nonzero digit?

(a) Evaluate \(\left(\begin{array}{c}11 \\ 4\end{array}\right)\). (b) Evaluate \(\left(\begin{array}{c}11 \\ 7\end{array}\right)\).

Evaluate \(\lim _{n \rightarrow \infty} n\left(\ln \left(3+\frac{1}{n}\right)-\ln 3\right)\).

Suppose \(x\) is a positive number. (a) Explain why \(x^{1 / n}=e^{(\ln x) / n}\) for every nonzero number \(n\). (b) Explain why $$ n\left(x^{1 / n}-1\right) \approx \ln x $$ if \(n\) is very large. (c) Explain why $$ \ln x=\lim _{n \rightarrow \infty} n\left(x^{1 / n}-1\right) $$ [A few books use the last equation above as the definition of the natural logarithm.]
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

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.