Infinite Geometric Sum Calculator
Infinite Sum (S∞): 0
Formula:
If |r| < 1, the infinite geometric sum is:
Example:
a = 5, r = 0.5
S∞ = 5 / (1 – 0.5) = 10
❗ Note: The formula only works if the absolute value of the ratio is less than 1 (|r| < 1).
If |r| < 1, the infinite geometric sum is:
S∞ = a / (1 - r)Example:
a = 5, r = 0.5
S∞ = 5 / (1 – 0.5) = 10
❗ Note: The formula only works if the absolute value of the ratio is less than 1 (|r| < 1).
🔢 Infinite Sum Calculator – Instantly Solve Infinite Series
The Infinite Sum Calculator helps you calculate the total of an infinite series — especially when the series converges to a finite value.
📘 What Is an Infinite Sum?
An infinite sum (or infinite series) adds terms endlessly: S=a1+a2+a3+a4+⋯S = a_1 + a_2 + a_3 + a_4 + \cdotsS=a1+a2+a3+a4+⋯
✅ If this sum approaches a specific value as more terms are added, it’s called a convergent series.
🔧 Features of the Infinite Sum Calculator
- Input a general term (like
1/n^2,(1/2)^n, etc.) - Choose the starting index (e.g.,
n = 1orn = 0) - Get:
- Sum of the series (if convergent)
- Step-by-step breakdown (in some tools)
- Convergence info (if diverges)
🧠 Common Series That Can Be Solved
| Type of Series | Example | Result |
|---|---|---|
| Geometric | ∑n=0∞(1/2)n\sum_{n=0}^{\infty} (1/2)^n∑n=0∞(1/2)n | 1 / (1 – 1/2) = 2 |
| p-Series | ∑n=1∞1/n2\sum_{n=1}^{\infty} 1/n^2∑n=1∞1/n2 | π² / 6 |
| Telescoping | ∑n=1∞1/n(n+1)\sum_{n=1}^{\infty} 1/n(n+1)∑n=1∞1/n(n+1) | 1 |
| Alternating | ∑(−1)n/n\sum (-1)^n/n∑(−1)n/n | ln(2) |
❓ FAQs – Infinite Sum Calculator
🔹 Can all infinite series be summed?
No — only convergent series have finite sums.
🔹 What is the sum of an infinite geometric series?
S=a1−r(if ∣r∣<1)S = \frac{a}{1 – r} \quad \text{(if } |r| < 1 \text{)}S=1−ra(if ∣r∣<1)
🔹 What if the series diverges?
The calculator will usually show that the sum is undefined or diverges.
🔹 Can this solve symbolic series like 1n(n+1)\frac{1}{n(n+1)}n(n+1)1?
Yes, advanced calculators like WolframAlpha or Symbolab can solve symbolic and telescoping series.