Sum to Infinity Calculator
Limit of a geometric series as n approaches infinity.
Formula first
Overview
This formula represents the limit of the sum of a geometric series as the number of terms approaches infinity. It defines the point of convergence for a sequence where each successive term is generated by multiplying the previous one by a constant ratio.
Symbols
Variables
= Sum to Infinity, a = First Term, r = Common Ratio
Apply it well
When To Use
When to use: This equation is applicable only to geometric progressions where the absolute value of the common ratio is strictly less than one. If the ratio is outside the range of -1 to 1, the series diverges and the sum is undefined.
Why it matters: Infinite sums are vital in financial modeling for determining the present value of perpetuities and in computer science for analyzing recursive algorithms. They also appear in physics when calculating the total distance traveled by bouncing objects or analyzing wave patterns.
Avoid these traps
Common Mistakes
- Using for r > 1.
- Forgetting condition |r| < 1.
One free problem
Practice Problem
A geometric series starts with 100 and each subsequent term is half of the one before it. Calculate the total sum of this series as it continues to infinity.
Solve for:
Hint: Identify the first term as 100 and the ratio as 0.5, then substitute into the formula.
The full worked solution stays in the interactive walkthrough.
References
Sources
- Wikipedia: Geometric series
- Britannica: Geometric series
- Stewart, James. Calculus: Early Transcendentals. Cengage Learning.
- AQA A-Level Mathematics — Pure (Sequences and Series)