Series Convergence Calculator
Determine if an infinite series converges or diverges using standard mathematical tests.
What is Series Convergence?
In mathematics, an infinite series is the sum of the terms of an infinite sequence. Convergence occurs when the sum of these terms approaches a specific, finite value as the number of terms increases toward infinity. If the sum grows without bound or fluctuates indefinitely, the series is said to diverge.
Geometric Series Test
A geometric series takes the form Σ a·rⁿ. It is one of the most common series types encountered in calculus and analysis.
– If |r| ≥ 1: The series diverges.
– Sum for convergent series: S = a / (1 – r)
P-Series Test
A P-series is a series of the form Σ 1/nᵖ, where n starts from 1. The behavior of this series depends entirely on the value of the exponent p.
– If p ≤ 1: The series diverges (e.g., the Harmonic Series where p=1).
Practical Examples
Example 1: Geometric Series
Consider a series where the first term is 5 and the ratio is 0.5. Since |0.5| is less than 1, the series converges. Using the formula 5 / (1 – 0.5), we find the sum is 10.
Example 2: P-Series
Consider the series Σ 1/n². Here, p = 2. Since 2 is greater than 1, this series converges. This specific series converges to π²/6.
Why Convergence Matters
Understanding convergence is critical in physics, engineering, and finance. It allows professionals to determine if complex functions can be represented by power series (like Taylor series) and ensures that numerical approximations will eventually stabilize at a correct value rather than spiraling into infinite error.