Last updated on August 5th, 2025
In mathematics, an infinite geometric series is a sum of an infinite sequence of terms where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In this topic, we will learn the formula for calculating the sum of an infinite geometric series.
The infinite geometric series has a sum that can be calculated if the series converges. Let’s learn the formula to calculate the sum of an infinite geometric series.
The sum of an infinite geometric series is given by the formula: [ S = frac{a}{1 - r} ] where ( S ) is the sum of the series, ( a ) is the first term, and ( r ) is the common ratio.
This formula is valid only if the absolute value of ( r ) is less than 1 ((|r| < 1)).
In mathematics and various applications, the infinite geometric series formula is essential for understanding the behavior of sequences and series.
Here are some important aspects of the infinite geometric series:
Students often find math formulas tricky and confusing.
Here are some tips and tricks to master the infinite geometric series formula:
The infinite geometric series formula plays a significant role in various real-life applications.
Here are some examples:
Students often make errors when working with infinite geometric series. Here are some common mistakes and ways to avoid them:
Find the sum of the infinite geometric series with first term 3 and common ratio 0.5.
The sum is 6.
Using the formula ( S = frac{a}{1 - r} ), where ( a = 3 ) and ( r = 0.5 ):
S = frac{3}{1 - 0.5}
= frac{3}{0.5}
= 6
Calculate the sum of the infinite geometric series where the first term is 8 and the common ratio is 0.25.
The sum is 10.67.
Using the formula ( S = frac{a}{1 - r} ), where ( a = 8 ) and ( r = 0.25 ):
S = frac{8}{1 - 0.25}
= frac{8}{0.75}
= 10.67
What is the sum of an infinite geometric series with first term 5 and common ratio 0.1?
The sum is 5.56.
Using the formula ( S = frac{a}{1 - r} ), where ( a = 5 ) and ( r = 0.1 ):
S = frac{5}{1 - 0.1}
= frac{5}{0.9}
= 5.56
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
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