What is an Infinite Series?

An infinite series is the sum of the terms of an infinite sequence. Mathematically, it is represented as the sum of an as n goes from a starting value (usually 0 or 1) to infinity. Understanding whether a series converges (settles on a specific number) or diverges (grows to infinity) is a fundamental concept in calculus and mathematical analysis.

How to Use the Infinite Series Calculator

Using this tool is straightforward. Simply input the general term of your sequence using 'n' as the variable. For example, if you want to find the sum of a geometric series where each term is half of the previous one, you would enter (1/2)^n.

• Formula: Enter the mathematical expression using standard operators (+, -, *, /, ^).
• Start Index: The value of n where your summation begins.
• Terms to Show: This determines how many individual steps the calculator will display to help you visualize the sequence.

Convergence vs. Divergence

The primary goal when dealing with infinite series is determining convergence. A series converges if the sequence of its partial sums approaches a finite limit. Our calculator uses numerical approximation to estimate the behavior. Common tests include the Ratio Test, the Root Test, and the Integral Test. For instance, the P-series 1/n^p converges only if p > 1.

Frequently Asked Questions

Q: Can every infinite series be summed?
A: No. Many series diverge, meaning their sum grows without bound or oscillates, such as the harmonic series 1/n.

Q: What is a Geometric Series?
A: It is a series with a constant ratio between successive terms. If the absolute value of the ratio is less than 1, the series converges to a / (1 - r).