Infinite Sum Calculator
Calculate Infinite Series with Convergence Testing
Analyze infinite series convergence and calculate sums for geometric series, p-series, and other mathematical series with step-by-step explanations.
Calculate infinite series with convergence testing
Geometric Series
Converges when |r| < 1, diverges when |r| ≥ 1. Sum = a/(1-r) when convergent.
p-Series
Converges when p > 1, diverges when p ≤ 1. Famous examples include Basel problem (p=2, sum=π²/6).
Alternating Series
May converge even when the corresponding positive series diverges, tested using the Alternating Series Test.
Supported Series Types
Convergent Series
- • Geometric Series: ∑ a·r^n (|r| < 1)
- • p-Series: ∑ 1/n^p (p > 1)
- • Alternating Series: ∑ (-1)^n·a_n
- • Basel Problem: π²/6
- • Exponential Series: e^x expansion
Convergence Tests
- • Ratio Test for geometric-like series
- • Comparison Test for p-series
- • Alternating Series Test
- • Built-in convergence analysis
- • Step-by-step explanations
Frequently Asked Questions
What is an infinite series?
An infinite series is the sum of infinitely many terms. The key question is whether this infinite sum converges to a finite value or diverges to infinity.
How do I know if a series converges?
For geometric series, convergence occurs when |r| < 1. For p-series, convergence occurs when p > 1. Our calculator automatically tests convergence for supported series types.
What's the difference between convergent and divergent series?
A convergent series approaches a finite sum as you add more terms. A divergent series either grows without bound or oscillates without settling on a value.
Can I calculate partial sums?
Yes! The calculator shows partial sums (first n terms) to help visualize how the series approaches its limit, along with the theoretical infinite sum for convergent series.
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