Summation Calculator

Q: What functions can I use in the expression?

You can use sin , cos , tan , asin , acos , atan , sinh , cosh , tanh , ln (natural log), log (base 10 log), exp , sqrt , abs (absolute value), and fact (factorial). You can also use the constants pi and e .

Q: How do I type exponents in the expression?

Use the ^ symbol. For example, type n^2 to square n, or n^3 to cube n. You can also press the x^y button on the keypad.

Q: What happens if the lower limit is bigger than the upper limit?

The calculator returns 0 . This is called an empty sum. There are no terms to add when the starting number is higher than the ending number.

Q: Does it matter which index variable I pick?

No. The letters n , i , k , and j all work the same way. Just make sure the variable in your expression matches the one you selected in the dropdown.

Q: How do I type multiplication?

Use the * symbol or press the × button on the keypad. You can also skip it in some cases. For example, 2n and 2*n both mean 2 times n.

Q: Why is my sum showing as infinity?

The sum grew too large for the calculator to handle. This happens with expressions that produce very big numbers over a wide range, like n^n or fact(n) with a high upper limit. Try using a smaller range.

Q: Can I sum a constant number instead of using the index variable?

Yes. If you type a number like 5 as the expression, the calculator adds that same number at every step. For example, summing 5 from n = 1 to n = 10 gives 5 × 10 = 50.

Q: How do I calculate the sum of 1/n (harmonic series)?

Set the expression to 1/n , the lower limit to 1 , and the upper limit to your chosen number. For example, with an upper limit of 10, the result is 1 + 1/2 + 1/3 + ... + 1/10 ≈ 2.9290.

Summation Calculator

Introduction

This summation calculator adds up a series of numbers using sigma notation. You pick a starting value, an ending value, and a math expression. The calculator then plugs in each number, solves the expression, and adds all the results together. It shows you the final sum, a step-by-step breakdown, and a chart of how the total grows.

Sigma notation is a short way to write long sums in math. Instead of writing out every single term, you use the Greek letter Σ (sigma) to say "add up all of these." For example, the sum of all whole numbers from 1 to 100 is written as Σ n, where n goes from 1 to 100. This tool does the calculation for you.

You can use basic operations like addition, subtraction, multiplication, division, and exponents. You can also use built-in functions like sin, cos, ln, sqrt, and factorial. Choose your index variable, set your limits, type your expression, and press Calculate. The calculator handles up to one million terms and gives results with up to 13 decimal places of precision.

How to Use Our Summation Calculator

Enter the parts of your sigma notation below and this calculator will add up all the terms for you. It gives you the total sum, a step-by-step solution, and a chart of the results.

Upper Limit: Type the number where the sum stops. This is the last value the index will reach.

Index Variable: Pick the letter you want to use in your expression. You can choose n, i, k, or j.

Lower Limit: Type the number where the sum starts. This is the first value the index will take.

Expression f(n): Type the formula that gets calculated at each step. You can use numbers, your index variable, operators like + − * / ^, and functions like sin, cos, sqrt, ln, and fact. You can also use the on-screen keypad to build your expression.

Decimal Precision: Choose how many decimal places you want in your answer, from 0 to 13.

Output Mode: Pick "Final Result Only" to see just the answer, or pick "Step-by-Step" to also see a table with every term and running total.

Click Calculate to get your result. Click Reset to clear everything and start over.

What Is Summation?

Summation is a math operation that adds up a list of numbers that follow a pattern. Instead of writing out every single number and adding them one by one, mathematicians use a shorthand symbol called sigma (Σ). This symbol tells you to add up a series of values from a starting point to an ending point.

How Sigma Notation Works

A summation has three main parts. The lower limit is the number where you start counting. The upper limit is the number where you stop. The expression is the rule that tells you what to do with each number before you add it. For example, if you sum n from 1 to 5, you get 1 + 2 + 3 + 4 + 5 = 15. If you sum n² from 1 to 5, you square each number first and then add: 1 + 4 + 9 + 16 + 25 = 55.

The Summation Formula

The general form of a summation is written as:

Σ f(n), where n goes from the lower limit to the upper limit.

The index variable n takes on every whole number value between the two limits. At each step, you plug n into the expression, calculate the result, and add it to a running total. The final total is the value of the summation.

Common Uses of Summation

Summation shows up in many areas of math and science. It is used to find the sum of arithmetic sequences, calculate areas under curves using integration in calculus, and work with series and sequences in algebra. Scientists use it in statistics to compute averages and standard deviations. Programmers use it in loops and algorithms. Any time you need to add up a long list of values that follow a rule, summation is the right tool.

Important Terms

Index variable – The letter (like n, i, or k) that counts through each step.
Lower limit – The first value the index takes.
Upper limit – The last value the index takes.
Term – The value you get after plugging the index into the expression at each step.
Running sum – The total so far as you add each term one at a time.

Tips for Working With Summations

If the lower limit is greater than the upper limit, the sum is 0 because there are no terms to add. This is called an empty sum. When you work with large ranges, the sum can grow very fast, especially with expressions like n² or n³. Expressions like 1/n grow slowly and are tied to the famous harmonic series in calculus. Knowing how your expression behaves helps you predict whether the sum will be small, large, or infinite.

Formulas used

Summation (Definition)

S = \sum_{n=a}^{b} f(n) = f(a) + f(a+1) + f(a+2) + \cdots + f(b)

Number of Terms

N = b - a + 1

Running (Cumulative) Sum

S_k = S_{k-1} + f(k), \quad S_{a-1} = 0

Empty Sum Convention

\sum_{n=a}^{b} f(n) = 0 \quad \text{if } b < a

Factorial (integer)

n! = \prod_{k=1}^{n} k

Factorial via Gamma (non-integer)

n! = \Gamma(n+1), \quad \Gamma(z) = \sqrt{2\pi}\; t^{\,z-\frac{1}{2}}\; e^{-t} \sum_{i=0}^{g+1} \frac{C_i}{z+i}, \quad t = z + g + \tfrac{1}{2}

Frequently asked questions

What functions can I use in the expression?

You can use sin, cos, tan, asin, acos, atan, sinh, cosh, tanh, ln (natural log), log (base 10 log), exp, sqrt, abs (absolute value), and fact (factorial). You can also use the constants pi and e.

How do I type exponents in the expression?

Use the ^ symbol. For example, type n^2 to square n, or n^3 to cube n. You can also press the x^y button on the keypad.

What happens if the lower limit is bigger than the upper limit?

The calculator returns 0. This is called an empty sum. There are no terms to add when the starting number is higher than the ending number.

Does it matter which index variable I pick?

No. The letters n, i, k, and j all work the same way. Just make sure the variable in your expression matches the one you selected in the dropdown.

How do I type multiplication?

Use the * symbol or press the × button on the keypad. You can also skip it in some cases. For example, 2n and 2*n both mean 2 times n.

Why is my sum showing as infinity?

The sum grew too large for the calculator to handle. This happens with expressions that produce very big numbers over a wide range, like n^n or fact(n) with a high upper limit. Try using a smaller range.

Can I sum a constant number instead of using the index variable?

Yes. If you type a number like 5 as the expression, the calculator adds that same number at every step. For example, summing 5 from n = 1 to n = 10 gives 5 × 10 = 50.

How do I calculate the sum of 1/n (harmonic series)?

Set the expression to 1/n, the lower limit to 1, and the upper limit to your chosen number. For example, with an upper limit of 10, the result is 1 + 1/2 + 1/3 + ... + 1/10 ≈ 2.9290.

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