Sum of Series Calculator
Calculate the sum of arithmetic, geometric, or power series. Supports both finite partial sums and infinite series (when convergent).
This free online sum of series calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.
First term of the series
Common difference (arith), ratio (geo), or exponent (power)
Number of terms to sum
Results
How to Use This Calculator
Enter your input values
Fill in all required input fields for the Sum of Series Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.
Review your inputs
Double-check that all values are correct and that you have selected the right units 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.
Read the results
The Sum of Series Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.
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.
Formula Reference
Sum of Series Calculator Formula
See calculator inputs for the governing equation
Variables: All variables and their units are labeled in the calculator interface above. Input fields accept values in multiple unit systems — select your preferred unit from the dropdown next to each field.
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.
About This Calculator
The Sum of Series Calculator is a free mathematical calculation tool for students, educators, and professionals who need quick, reliable results. Calculate the sum of arithmetic, geometric, or power series. Supports both finite partial sums and infinite series (when convergent). The underlying algorithms implement well-established mathematical formulas and numerical methods. Results are computed instantly in the browser. This tool is useful for learning, verification of hand calculations, and rapid exploration of mathematical relationships. All computation happens locally — no data is sent to a server.
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
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.
S₁₀₀ = 5050
Example 2: Geometric Series Sum
Find the sum of 3 + 6 + 12 + … for 8 terms.
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
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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.