Skip to main content
Ch. 8 - Sequences, Induction, and Probability
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 8 - Sequences, Induction, and ProbabilityProblem 41
Chapter 9, Problem 41

Find the sum of each infinite geometric series. 1 - 1/2 + 1/4 - 1/8 + ...

Verified step by step guidance
1
Identify the first term \( a \) of the infinite geometric series. In this series, the first term is \( a = 1 \).
Determine the common ratio \( r \) by dividing the second term by the first term: \( r = \frac{-\frac{1}{2}}{1} = -\frac{1}{2} \).
Check if the series converges by verifying that the absolute value of the common ratio is less than 1: \( |r| = \left| -\frac{1}{2} \right| = \frac{1}{2} < 1 \), so the series converges.
Use the formula for the sum \( S \) of an infinite geometric series: \[ S = \frac{a}{1 - r} \].
Substitute the values of \( a \) and \( r \) into the formula to express the sum: \[ S = \frac{1}{1 - \left(-\frac{1}{2}\right)} \].

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
4m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Infinite Geometric Series

An infinite geometric series is the sum of infinitely many terms where each term is found by multiplying the previous term by a constant ratio. The series continues indefinitely, and its behavior depends on the common ratio's value.
Recommended video:
Guided course
4:18
Geometric Sequences - Recursive Formula

Common Ratio

The common ratio is the factor by which each term in a geometric series is multiplied to get the next term. It is found by dividing any term by its preceding term. For convergence of an infinite series, the absolute value of the common ratio must be less than 1.
Recommended video:
5:57
Graphs of Common Functions

Sum Formula for Infinite Geometric Series

If the absolute value of the common ratio r is less than 1, the sum S of the infinite geometric series with first term a is given by S = a / (1 - r). This formula allows calculation of the series' sum without adding infinitely many terms.
Recommended video:
Guided course
4:18
Geometric Sequences - Recursive Formula
Related Practice
Textbook Question

Find the sum of each infinite geometric series. ∑i=1∞8(−0.3)i−1\(\sum\)_{i=1}^{\(\infty\)} 8(-0.3)^{i-1}

1
views
Textbook Question

Express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. 12+22+32+⋯+1521^2+2^2+3^2+⋯+ 15^2

Textbook Question

Find the sum of the first 60 positive even integers.

1
views
Textbook Question

Find each indicated sum. ∑i=15i!(i−1)!\(\sum\)_{i=1}^{5} \(\frac{i!}{(i-1)!}\)

1
views
Textbook Question

Use the formula for the sum of the first n terms of a geometric sequence to find the indicated sum.

∑i=165i\(\sum\)_{i=1}^{6} 5^i

Textbook Question

Find the sum of the first 15 terms of the geometric sequence: 5, -15, 45, -135