Skip to main content
math

Sum of Series Calculator

Calculate the sum of arithmetic, geometric, or power series. Supports both finite partial sums and infinite series (when convergent).

By Christopher FloiedPublished Updated

First term of the series

Common difference (arith), ratio (geo), or exponent (power)

Minimum: 1. Whole numbers only

Number of terms to sum

Results

Sum

100

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the Sum of Series Calculator. Follow the units and input format printed next to each field. Where a unit selector is available, choose the units that match your data.

2

Review your inputs

Double-check that all values are correct and match the units or format shown for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The Sum of Series Calculator computes the output in your browser and displays the results, with units clearly labeled, as soon as they are ready — no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

When to Use This Calculator

  • •Use the Sum of Series Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
  • •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
  • •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
  • •Use it to explore the behavior of mathematical functions across a range of inputs.

Related Calculators

Geometric Sequence Calculator

Calculate the nth term, partial sum, and infinite sum of a geometric sequence defined by its first term and common ratio.

Derivative Calculator

Differentiate sums of power terms such as 3x² + 5x − 7, term by term with the power rule.

Exponent Calculator

Calculate the value of a base raised to a power (bⁿ). Supports integer, fractional, and negative exponents.

Sequence Calculator

Evaluate a general sequence defined by an explicit formula a(n) for a range of term numbers. Identify patterns and compute partial sums.

Adding and Subtracting Polynomials Calculator

Add or subtract two polynomials by combining like terms. Supports up to degree 4 polynomials.

AND Calculator

Perform the bitwise AND operation on two numbers. Each output bit is 1 only when both corresponding input bits are 1.

About Sum of Series Calculator

A series is the sum of the terms of a sequence. Series are central to calculus, analysis, and applied mathematics. Finite series compute definite sums, while infinite series extend this concept to infinitely many terms — converging to a limit when certain conditions are met. This calculator handles three fundamental types: arithmetic series (constant difference between terms), geometric series (constant ratio between terms), and power series (sums of k raised to a fixed power). Mastering series is essential for understanding integration (Riemann sums), Taylor expansions, Fourier analysis, probability distributions, and financial mathematics. The formulas for these sums have been known for centuries — Gauss's trick for arithmetic sums, Euclid's formula for geometric sums, and Faulhaber's formulas for power sums — and they remain among the most useful tools in a mathematician's arsenal.

The Math Behind It

An arithmetic series sums terms with constant difference d: S = n/2 × (first + last). A geometric series sums terms with constant ratio r: S = a₁(1−rⁿ)/(1−r). Power sums Σk^p for p = 1, 2, 3 have well-known closed forms. For p = 1: n(n+1)/2. For p = 2: n(n+1)(2n+1)/6. For p = 3: [n(n+1)/2]². General Faulhaber formulas express Σk^p as a polynomial of degree p+1 in n involving Bernoulli numbers. Convergence of infinite series requires the terms to approach zero (necessary but not sufficient). The ratio test, root test, comparison test, and integral test are standard tools for determining convergence. Absolute convergence (Σ|aₙ| converges) implies convergence, but conditional convergence allows rearrangement to change the sum (Riemann rearrangement theorem). Power series Σaₙxⁿ converge within a radius of convergence R and define analytic functions.

Formula Reference

Arithmetic Series

Sₙ = n/2 × (2a₁ + (n−1)d)

Variables: a₁ = first term, d = common difference

Geometric Series

Sₙ = a₁(1 − rⁿ)/(1 − r)

Variables: r = common ratio, r ≠ 1

Sum of Squares

Σk² = n(n+1)(2n+1)/6

Variables: k = 1 to n

Sum of Cubes

Σk³ = [n(n+1)/2]²

Variables: k = 1 to n

Worked Examples

Example 1: Sum of First 100 Positive Integers

Find 1 + 2 + 3 + … + 100.

Step 1:This is an arithmetic series with a₁ = 1, d = 1, n = 100
Step 2:S₁₀₀ = 100/2 × (2(1) + 99(1))
Step 3:S₁₀₀ = 50 × 101

S₁₀₀ = 5050

Example 2: Geometric Series Sum

Find the sum of 3 + 6 + 12 + … for 8 terms.

Step 1:a₁ = 3, r = 2, n = 8
Step 2:S₈ = 3(1 − 2⁸)/(1 − 2)
Step 3:S₈ = 3(1 − 256)/(−1) = 3 × 255

S₈ = 765

Common Mistakes & Tips

  • !Using the geometric sum formula when r = 1 — this causes division by zero; use S = n × a₁ instead.
  • !Confusing the number of terms n with the last term value.
  • !Forgetting that an infinite geometric series only converges when |r| < 1.
  • !Applying power sum formulas with the wrong exponent.

Related Concepts

Arithmetic Sequence

The underlying sequence whose terms are summed in an arithmetic series.

Geometric Sequence

The underlying sequence for geometric series.

Harmonic Number

The partial sums of the harmonic series 1 + 1/2 + 1/3 + ….

Used in These Calculators

Calculators that build on or apply the concepts from this page:

Arithmetic Sequence Calculator

Convolution Calculator

Geometric Sequence Calculator

Harmonic Number Calculator

Lagrange Error Bound Calculator

Sequence Calculator

Sum of a Linear Number Sequence Calculator

Frequently Asked Questions

What is the difference between a sequence and a series?

A sequence is an ordered list of numbers; a series is the sum of those numbers. For example, 1, 2, 3 is a sequence; 1 + 2 + 3 = 6 is a series.

Can a series with terms that don't approach zero converge?

No. If the terms aₙ do not approach zero, the series must diverge. However, terms approaching zero is necessary but not sufficient — the harmonic series 1 + 1/2 + 1/3 + … diverges despite its terms approaching zero.

Add the live Sum of Series Calculator to your site. Free to embed, with a link back to MegaCalc.