Find The Sum of The Following Infinite Series Calculator

This calculator helps you find the sum of common infinite series. Whether you're studying calculus or need to solve a practical problem, this tool provides quick, accurate results with step-by-step explanations.

How to Use This Calculator

Using our infinite series calculator is simple:

Select the type of infinite series you want to calculate from the dropdown menu.
Enter the required values in the input fields (such as the first term and common ratio for geometric series).
Click the "Calculate" button to see the sum of the series.
Review the result and the step-by-step solution provided.

The calculator supports several common types of infinite series, including geometric series, p-series, and alternating series. Each type has specific conditions for convergence.

Common Infinite Series

Infinite series are sequences of numbers that extend infinitely. The sum of an infinite series is the limit of the sum of its partial sums. Not all infinite series converge to a finite limit; some diverge to infinity.

Geometric Series

A geometric series has the form:

S = a + ar + ar² + ar³ + ... = a / (1 - r), for |r| < 1

Where:

a is the first term
r is the common ratio between terms

The series converges if the absolute value of r is less than 1.

P-Series

A p-series has the form:

S = 1 + 1/2 + 1/3 + 1/4 + ... + 1/n + ...

This series converges if p > 1 and diverges if p ≤ 1.

Alternating Series

An alternating series has terms that alternate in sign. The Leibniz test can be used to determine convergence.

Calculation Method

The calculator uses the following formulas to compute the sum of infinite series:

Geometric Series

S = a / (1 - r) if |r| < 1

P-Series

S = ∞ if p ≤ 1 S = ζ(p) if p > 1 (where ζ is the Riemann zeta function)

Alternating Series

S = Σ (-1)^(n+1) * b_n where b_n is a decreasing sequence of positive numbers

The calculator checks for convergence conditions before computing the sum. If the series does not converge, it will indicate that the sum is infinite.

Worked Examples

Example 1: Geometric Series

Find the sum of the infinite series: 2 + 1 + 0.5 + 0.25 + ...

This is a geometric series with a = 2 and r = 0.5.

Using the formula:

S = 2 / (1 - 0.5) = 4

The sum of the series is 4.

Example 2: P-Series

Find the sum of the infinite series: 1 + 1/2 + 1/3 + 1/4 + ...

This is a p-series with p = 1.

Since p ≤ 1, the series diverges to infinity.

Example 3: Alternating Series

Find the sum of the infinite series: 1 - 1/2 + 1/3 - 1/4 + ...

This is an alternating series that converges to ln(2) ≈ 0.6931.

Frequently Asked Questions

What types of infinite series can this calculator solve?

This calculator can solve geometric series, p-series, and alternating series. Each type has specific conditions for convergence.

How do I know if an infinite series converges?

The calculator checks convergence conditions for each type of series. For geometric series, it checks if the absolute value of the common ratio is less than 1. For p-series, it checks if the exponent is greater than 1.

Can I use this calculator for financial calculations?

While this calculator is primarily designed for mathematical series, the geometric series formula can be applied to certain financial calculations involving annuities or present values.

What if the series doesn't converge?

If the series does not converge, the calculator will indicate that the sum is infinite. This means the partial sums grow without bound as more terms are added.