Infinite Series Formulas
Learn key infinite series formulas including geometric, harmonic, and p-series.
Understand convergence tests with worked examples.
The Formulas
P-series: Σ 1/nᵖ converges if p > 1
Harmonic: Σ 1/n diverges (p = 1)
An infinite series is the sum of infinitely many terms. The central question is whether the series converges (approaches a finite value) or diverges (grows without bound or oscillates).
The geometric series is the most fundamental infinite series. It converges to a/(1-r) when the common ratio r has absolute value less than 1. For example, 1 + 1/2 + 1/4 + 1/8 + ... = 2.
The harmonic series 1 + 1/2 + 1/3 + 1/4 + ... famously diverges, even though the terms approach zero. This was first proven by Nicole Oresme in the 14th century. The p-series generalizes this: Σ 1/nᵖ converges when p > 1 and diverges when p ≤ 1.
Several convergence tests exist: the ratio test, root test, comparison test, integral test, and alternating series test. Choosing the right test depends on the form of the series.
Variables
| Symbol | Meaning |
|---|---|
| a | First term of the series |
| r | Common ratio (for geometric series) |
| n | Index of summation (typically starts at 1 or 0) |
| p | Exponent in a p-series |
| Σ | Summation — adding up all terms from n = 1 to infinity |
Example 1
Find the sum of the infinite geometric series: 5 + 5/3 + 5/9 + 5/27 + ...
Identify: a = 5, r = 1/3
Check convergence: |r| = 1/3 < 1, so the series converges
Apply the formula: S = a/(1-r) = 5/(1 - 1/3) = 5/(2/3)
S = 15/2 = 7.5
Example 2
Does the series Σ 1/n² (from n=1 to infinity) converge? If so, what is its value?
This is a p-series with p = 2. Since p > 1, the series converges.
This famous series was solved by Leonhard Euler in 1735 (the Basel problem).
Σ 1/n² = 1 + 1/4 + 1/9 + 1/16 + ...
Σ 1/n² = π²/6 ≈ 1.6449
When to Use It
Infinite series appear throughout mathematics, physics, and engineering.
- Computing values of transcendental functions (e, π, trigonometric values)
- Signal processing and Fourier analysis
- Financial calculations involving perpetuities
- Probability theory and generating functions