How do I write a summation here?

Read the sigma notation onto the five fields. What follows the sigma goes in Expression, the letter under it goes in Index, and the two bounds go in From and To. Step is 1 unless you are skipping values. Powers use ^, so i^2 from 1 to 10 gives 385, and multiplication can be written 2*i or just 2i. Brackets work as you expect, pi and e are available, and the named functions are abs, acos, asin, atan, cbrt, ceil, comb, cos, cosh, exp, fact, floor, ln, log, log10, log2, max, min, mod, pow, round, sign, sin, sinh, sqrt, tan and tanh. Here log is base 10 and ln is natural. A factorial can be written i! or fact(i), and comb(10, i) is the binomial coefficient, so summing comb(10, i) from 0 to 10 gives 1,024.

What is the sum of the numbers from 1 to 100?

5,050. That is the opening sum on the page: i from 1 to 100 with a step of 1. Because the terms rise by the same amount every time, the page uses the closed form rather than adding one hundred numbers, and it prints the working next to the answer: 100 × (1 + 100) / 2. The same shortcut covers recognized linear expressions, so 2*i - 1 from 1 to 10 comes back as 100 with 10 × (1 + 19) / 2 beside it. Equivalent expressions such as round(i) use term-by-term addition instead.

Can it sum an infinite series?

No, and it does not pretend to. It adds a finite number of terms between the bounds you give, and an upper bound of inf or infinity is refused rather than guessed at. What it will do is a partial sum, which is often what the question really is: 1/i^2 from 1 to 1,000 comes back as 1.64393456668 after 1,000 terms. That figure is the total of the terms it added, and the page makes no claim about what the series does past them.

Why does my sum say the range is empty?

Because a positive step can never walk from the lower bound down to a smaller upper bound, so there are no terms at all. That is a result rather than an error: the page reports an empty range and a sum of 0, and says which way the index was trying to go. If you meant to count down, set the step to -1. From 5 to 1 with a step of -1 is 5 + 4 + 3 + 2 + 1, which is 15.

The total is exact when every term is evaluated through verified integer operations and stays within 9,007,199,254,740,991 in magnitude. Whole-number literals and integer operations are checked; decimal or scientific literals and other functions can remain approximate even when their output looks whole. Recognized linear expressions with verified integer coefficients use an exact integer formula, including when their terms or total are larger. The cubes from 1 to 100,000 come to 25,000,500,002,500,000,000 here because every cube fits the safe-term range; a plain floating-point loop returns 25,000,500,002,500,145,000. Larger or already-rounded terms use a compensated floating-point total, which reduces rounding in additions but cannot recover precision lost when evaluating a term. Fractional results are shown to twelve significant digits. The line under the readout names the method used.

How many terms will it add?

Up to 100,000 in one sum. Up to 5,000 terms the total updates as you type; past that the stage shows the term count and waits for you to press Sum, and a Cancel button stops a long run while it is going. Ask for more than 100,000 terms and it says how many the range needs rather than quietly summing part of it. The step is how you cover a wider range: i from 1 to 1,000,000 in steps of 10 is 100,000 terms and runs, though it adds only the values the step lands on, which is a different sum from every whole number in that range.

Does it show the terms it added?

Yes, under the readout. Eight terms or fewer are printed in full, so 1 + 3 + 5 + 7 + 9 is there to check; a longer sum prints the first six and the last one, so i^2 from 1 to 10 reads 1 + 4 + 9 + 16 + 25 + 36 + ... + 100. The first and last terms also have their own cells in the readout, which is the quickest way to catch a bound that is one out.

How do I sum a finite geometric series?

Use a numeric first term and ratio in the expression. For first term 3 and ratio 2, enter 3*2^i from 0 to 4 with step 1: the five terms are 3 + 6 + 12 + 24 + 48, totaling 93. The finite geometric-series page opens on this example. Change the numbers for your own finite series; letters for unknown coefficients are not supported.

Does it show symbolic steps or find a formula from a list?

It shows the terms and the method used to evaluate your expression. Recognized linear expressions include the numeric substitution in n(first + last)/2. It does not infer an expression from a pasted list, simplify symbolic sums, shift an index algebraically, or perform nested sums with multiple indices.

Can I use this on a phone or in a spreadsheet?

It works in a current browser on a phone or computer, with no installation. In Excel or Google Sheets, place the index values in column A, calculate each term in column B, then use SUM over those cells. For example, =A1^2 calculates a square and =SUM(B1:B10) adds ten such cells. Spreadsheet numeric precision can differ from the integer total shown here.

Summation Calculator

Calculate a finite sigma sum by entering an expression, its index, two integer bounds and a step. The tool below shows the total, term count, first and last terms, and the method used, so an off-by-one bound is visible. It works in your browser with no account and nothing uploaded.

Sum5,050i from 1 to 100, step 1 · 100 terms · first 1, last 100

Expression

Index

From

Step

Sum

5,050

Terms

100

First term

Last term

100

1 + 2 + 3 + 4 + 5 + 6 + ... + 100

Arithmetic sequence, so the closed form n(first + last)/2 gives it in one step: 100 × (1 + 100) / 2. The terms are whole numbers, so the total is exact.

Write powers with ^, multiply with * or by writing 2i, and use brackets freely. Functions: abs, acos, asin, atan, cbrt, ceil, comb, cos, cosh, exp, fact, floor, ln, log, log10, log2, max, min, mod, pow, round, sign, sin, sinh, sqrt, tan, tanh, plus the constants pi and e. Here log is base 10 and ln is natural.

i from 1 to 100step 1
5,050
i^2 from 1 to 10step 1
385
Upper bound below lower5 down to 1
empty range, sum 0

What this is exact about. Exact for integer terms whose evaluation is verified within the safe-integer range (up to 9,007,199,254,740,991 in magnitude); integer totals can be larger. Recognized linear expressions with verified integer coefficients also use exact integer arithmetic. Larger or already-rounded terms use floating-point approximation, and a compensated running sum reduces rounding in additions. It evaluates a finite number of terms, capped at 100,000 with a cancel button, and the closed form is used where the sequence is arithmetic and its linear structure is recognized. It makes no claim about whether an infinite series converges.

Common questions

How do I write a summation here?: Read the sigma notation onto the five fields. What follows the sigma goes in Expression, the letter under it goes in Index, and the two bounds go in From and To. Step is 1 unless you are skipping values. Powers use ^, so i^2 from 1 to 10 gives 385, and multiplication can be written 2*i or just 2i. Brackets work as you expect, pi and e are available, and the named functions are abs, acos, asin, atan, cbrt, ceil, comb, cos, cosh, exp, fact, floor, ln, log, log10, log2, max, min, mod, pow, round, sign, sin, sinh, sqrt, tan and tanh. Here log is base 10 and ln is natural. A factorial can be written i! or fact(i), and comb(10, i) is the binomial coefficient, so summing comb(10, i) from 0 to 10 gives 1,024.
What is the sum of the numbers from 1 to 100?: 5,050. That is the opening sum on the page: i from 1 to 100 with a step of 1. Because the terms rise by the same amount every time, the page uses the closed form rather than adding one hundred numbers, and it prints the working next to the answer: 100 × (1 + 100) / 2. The same shortcut covers recognized linear expressions, so 2*i - 1 from 1 to 10 comes back as 100 with 10 × (1 + 19) / 2 beside it. Equivalent expressions such as round(i) use term-by-term addition instead.
Can it sum an infinite series?: No, and it does not pretend to. It adds a finite number of terms between the bounds you give, and an upper bound of inf or infinity is refused rather than guessed at. What it will do is a partial sum, which is often what the question really is: 1/i^2 from 1 to 1,000 comes back as 1.64393456668 after 1,000 terms. That figure is the total of the terms it added, and the page makes no claim about what the series does past them.
Why does my sum say the range is empty?: Because a positive step can never walk from the lower bound down to a smaller upper bound, so there are no terms at all. That is a result rather than an error: the page reports an empty range and a sum of 0, and says which way the index was trying to go. If you meant to count down, set the step to -1. From 5 to 1 with a step of -1 is 5 + 4 + 3 + 2 + 1, which is 15.
Is the answer exact?: The total is exact when every term is evaluated through verified integer operations and stays within 9,007,199,254,740,991 in magnitude. Whole-number literals and integer operations are checked; decimal or scientific literals and other functions can remain approximate even when their output looks whole. Recognized linear expressions with verified integer coefficients use an exact integer formula, including when their terms or total are larger. The cubes from 1 to 100,000 come to 25,000,500,002,500,000,000 here because every cube fits the safe-term range; a plain floating-point loop returns 25,000,500,002,500,145,000. Larger or already-rounded terms use a compensated floating-point total, which reduces rounding in additions but cannot recover precision lost when evaluating a term. Fractional results are shown to twelve significant digits. The line under the readout names the method used.
How many terms will it add?: Up to 100,000 in one sum. Up to 5,000 terms the total updates as you type; past that the stage shows the term count and waits for you to press Sum, and a Cancel button stops a long run while it is going. Ask for more than 100,000 terms and it says how many the range needs rather than quietly summing part of it. The step is how you cover a wider range: i from 1 to 1,000,000 in steps of 10 is 100,000 terms and runs, though it adds only the values the step lands on, which is a different sum from every whole number in that range.
Does it show the terms it added?: Yes, under the readout. Eight terms or fewer are printed in full, so 1 + 3 + 5 + 7 + 9 is there to check; a longer sum prints the first six and the last one, so i^2 from 1 to 10 reads 1 + 4 + 9 + 16 + 25 + 36 + ... + 100. The first and last terms also have their own cells in the readout, which is the quickest way to catch a bound that is one out.
How do I sum a finite geometric series?: Use a numeric first term and ratio in the expression. For first term 3 and ratio 2, enter 3*2^i from 0 to 4 with step 1: the five terms are 3 + 6 + 12 + 24 + 48, totaling 93. The finite geometric-series page opens on this example. Change the numbers for your own finite series; letters for unknown coefficients are not supported.
Does it show symbolic steps or find a formula from a list?: It shows the terms and the method used to evaluate your expression. Recognized linear expressions include the numeric substitution in n(first + last)/2. It does not infer an expression from a pasted list, simplify symbolic sums, shift an index algebraically, or perform nested sums with multiple indices.
Can I use this on a phone or in a spreadsheet?: It works in a current browser on a phone or computer, with no installation. In Excel or Google Sheets, place the index values in column A, calculate each term in column B, then use SUM over those cells. For example, =A1^2 calculates a square and =SUM(B1:B10) adds ten such cells. Spreadsheet numeric precision can differ from the integer total shown here.

Exact for integer terms whose evaluation is verified within the safe-integer range (up to 9,007,199,254,740,991 in magnitude); integer totals can be larger. Recognized linear expressions with verified integer coefficients also use exact integer arithmetic. Larger or already-rounded terms use floating-point approximation, and a compensated running sum reduces rounding in additions. It evaluates a finite number of terms, capped at 100,000 with a cancel button, and the closed form is used where the sequence is arithmetic and its linear structure is recognized. It makes no claim about whether an infinite series converges.