About the Series Sum Calculator
A series is the sum of the terms of a sequence. In an arithmetic series each term differs from the last by the same amount, and in a geometric series each term is a fixed multiple of the last. This calculator gives the sum of the first n terms for either kind, along with the last term. For a geometric series with a ratio between −1 and 1 it also gives the sum to infinity. Type the kind of series, the first term, the common difference or ratio and the number of terms and the series sum appears at once, with a few related figures underneath, and you can copy the result with one tap. The sums run in your browser, so nothing you enter is uploaded.
Updated October 2026
How to use the Series Sum Calculator
- 1Enter the kind of series, the first term, the common difference or ratio and the number of terms.
- 2Pick the units or options from the menus if the series sum tool offers them.
- 3Read the series sum in the large result box.
- 4Look at the extra rows for related figures.
- 5Press Copy result to keep the answer.
When you need the series sum
Students use these sums in algebra and calculus. Everyday uses include savings plans that grow by a fixed amount each month, loan repayments, bouncing balls and any process where each step is a fixed fraction of the one before.
The arithmetic sum of the numbers 1 to 100 is the famous 5,050. The geometric series with ratio one half adds up to a finite total even with infinitely many terms, which resolves paradoxes like Zeno's.
How the series sum is worked out
Arithmetic: Sₙ = n ÷ 2 × (2a + (n − 1)d). Geometric: Sₙ = a × (1 − rⁿ) ÷ (1 − r), and the infinite sum is a ÷ (1 − r) when |r| < 1. The arithmetic series with first term 2, difference 3 and 10 terms ends at 29 and adds up to 155. The geometric series 3, 6, 12, 24, 48 has the sum 93. The series 1 + 1/2 + 1/4 + … with 10 terms adds to 1.998 and, taken forever, to exactly 2.
Every answer on this page comes from that one rule, so you can check any result by hand with a calculator. The figures shown underneath, such as other units, are conversions of the same answer.
- Arithmetic and geometric
- Last term shown
- Infinite sum when it exists
Mistakes to avoid with series sum
The number of terms must be a whole number. For a geometric series with a ratio of 1 or more in size, the infinite sum does not exist, and the tool says so. Very large ratios or term counts can produce numbers too big to show.
On phone and computer
On a phone, type into the boxes one after another and read the series sum underneath; Copy result puts it on the clipboard for a message or a note. On a computer the inputs sit on the left and the answer on the right, so you can change one figure and watch the series sum move. The series sum calculator works the same on both. Series come from patterns, so related tools help. Build a pattern with the Number Sequence Calculator, find Fibonacci terms with the Fibonacci Calculator, and see repeated growth with the Exponential Growth Calculator.
Features
- Whole number of terms
- |r| < 1 for infinite sums
- Check which kind you have
- Clear messages when the series sum inputs make no sense
- Copy the result with one tap
- Private: nothing is uploaded
Frequently asked questions
How do I calculate series sum?
Arithmetic: Sₙ = n ÷ 2 × (2a + (n − 1)d). Geometric: Sₙ = a × (1 − rⁿ) ÷ (1 − r), and the infinite sum is a ÷ (1 − r) when |r| < 1. The arithmetic series with first term 2, difference 3 and 10 terms ends at 29 and adds up to 155. The geometric series 3, 6, 12, 24, 48 has the sum 93. The series 1 + 1/2 + 1/4 + … with 10 terms adds to 1.998 and, taken forever, to exactly 2.
What is the formula for series sum?
Arithmetic: Sₙ = n ÷ 2 × (2a + (n − 1)d). Geometric: Sₙ = a × (1 − rⁿ) ÷ (1 − r), and the infinite sum is a ÷ (1 − r) when |r| < 1. The series sum calculator applies it exactly and rounds only the displayed answer to six significant figures.
What is the sum of the first 100 whole numbers?
5,050, from n(n + 1) ÷ 2 with n = 100.
When does an infinite geometric series have a sum?
Only when the absolute value of the ratio is less than 1.
How do I tell the two series apart?
If you add or subtract the same amount to get the next term, it is arithmetic. If you multiply by the same number, it is geometric.
Is this series sum calculator free and private?
Yes. It is free, needs no account and runs in your browser, so the numbers you type are not uploaded or stored.