Textbook Question
Find the sum of each infinite geometric series. 1 - 1/2 + 1/4 - 1/8 + ...
849
views
9. Sequences, Series, & Induction
Geometric Sequences
2:43 minutesProblem 47
Textbook Question
Find the sum of each infinite geometric series. 2 - 1 + 1/2 - 1/4 + ...
Verified step by step guidance1
Identify the first term \( a \) of the infinite geometric series. In this series, the first term is \( 2 \).
Determine the common ratio \( r \) by dividing the second term by the first term: \( r = \frac{-1}{2} \).
Check if the series converges by verifying that the absolute value of the common ratio is less than 1, i.e., \( |r| < 1 \).
Use the formula for the sum of an infinite geometric series, which is \( S = \frac{a}{1 - r} \), where \( a \) is the first term and \( r \) is the common ratio.
Substitute the values of \( a \) and \( r \) into the formula to express the sum \( S \) without calculating the final numerical value.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Infinite Geometric Series
An infinite geometric series is a 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.
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 and is crucial for determining the series' convergence.
Recommended video:
5:57Graphs of Common Functions
Sum of an Infinite Geometric Series
If the absolute value of the common ratio is less than 1, the infinite geometric series converges, and its sum can be calculated using the formula S = a / (1 - r), where 'a' is the first term and 'r' is the common ratio.
Recommended video:
Guided course
4:18Geometric Sequences - Recursive Formula
Related Videos
Related Practice