What does the summation symbol (sigma) mean?

The capital Greek letter sigma (S) followed by f(i) means "add up the values of f(i) as i takes every integer from the lower bound to the upper bound". For example, summing i^2 from i=1 to i=4 gives 1+4+9+16 = 30.

What is the formula for the sum of an arithmetic series?

An arithmetic series has a constant difference d between terms. The sum of n terms is S = n/2 * (first + last), equivalently S = n/2 * (2a + (n-1)d) where a is the first term. For example, 1+3+5+7+9 has a=1, d=2, n=5, so S = 5/2 * (1+9) = 25.

How do you sum a geometric series?

A geometric series multiplies each term by a constant ratio r. The finite sum of n terms starting at a is S = a * (1 - r^n) / (1 - r) when r is not equal to 1. When r = 1 every term equals a and S = a * n. An infinite geometric series converges only when the absolute value of r is less than 1, giving S = a / (1 - r).

What is the formula for the sum of the first n squares?

The sum of i^2 from i=1 to n equals n*(n+1)*(2n+1) / 6. For n=10 that is 10*11*21/6 = 385. The calculator verifies this automatically; you can check by choosing Power with p=2 and bounds 1 to 10.

Why does the harmonic series diverge?

The harmonic series 1 + 1/2 + 1/3 + ... grows without bound, just extremely slowly. Adding terms up to i=10 gives about 2.93; up to i=1000 gives about 7.49; up to i=1,000,000 gives about 14.4. Because terms never shrink fast enough, partial sums eventually exceed any fixed value. The calculator shows the partial sums so you can see the growth rate for yourself.

Can I sum a custom expression like 1/(i*(i+1))?

Yes. In Custom mode enter the expression using i for the index and ^ for powers. Parentheses, +, -, *, and / all work. The expression is evaluated safely in-browser without any network call. Note that 1/(i*(i+1)) has the elegant closed form 1 - 1/(n+1), which you can verify by comparing the calculator result with that formula.

Summation Calculator

The Summation Calculator evaluates finite series of the form

S = f(from) + f(from+1) + … + f(to)

instantly in your browser. Type any expression in i — powers, fractions, polynomials — or choose a built-in series (arithmetic, geometric, power, harmonic) and get the exact sum, every individual term, running partial sums, and a mini sparkline showing how the sum grows. Nothing is sent to a server.

How it works

For each integer value of i from the lower bound to the upper bound, the calculator evaluates your expression, adds the result to a running total, and records the partial sum at each step. You see the final total, the first and last terms, and the average term value.

For built-in series types it also shows the closed-form formula so you can cross-check by hand:

Arithmetic — S = n/2 * (first + last), where n is the number of terms.
Geometric — S = a * (1 - r^n) / (1 - r) for ratio r not equal to 1; or a * n when r = 1.
Power (i^p) — for p=1: n*(n+1)/2; for p=2: n*(n+1)(2n+1)/6; for p=3: [n(n+1)/2]^2.
Harmonic — no simple closed form; partial sums grow as ln(n) + 0.5772 (the Euler–Mascheroni constant) for large n.

The sparkline plots each partial sum S(1), S(2), …, S(n), making it easy to see whether the series is growing quickly, linearly, or levelling off.

Worked example

Sum the first 10 perfect squares: sum of i^2 from i=1 to i=10.

Using the closed form: 10 * 11 * 21 / 6 = 385.

i	i^2	Partial sum
1	1	1
2	4	5
3	9	14
4	16	30
5	25	55
6	36	91
7	49	140
8	64	204
9	81	285
10	100	385

The calculator confirms 385 and shows the closed-form derivation. Try it with the Power preset, exponent 2, bounds 1 to 10.

Sigma notation explained

Sigma notation is compact shorthand for “sum this expression over an index range”:

  10
  S   i^2   =   1^2 + 2^2 + 3^2 + ... + 10^2   =   385
 i=1

The letter below sigma (here i=1) is the starting index, the number above (10) is the ending index, and the expression to the right (i^2) is the term formula. Any symbol can be used for the index — i, k, j, n are all conventional.

Custom expression guide

In the Custom mode you can use:

i — the running index
^ — exponentiation (e.g. i^3)
+, -, *, / — arithmetic
( ) — grouping (e.g. 1/(i*(i+1)))

Examples worth trying: i^3 - i, 1/(2^i), (2*i-1) (odd numbers), i*(i+1)*(i+2)/6 (tetrahedral numbers). The evaluator is sandboxed and never reaches the network.

Formula reference

Series	nth term	Sum formula
Natural numbers	i	n*(n+1)/2
Squares	i^2	n(n+1)(2n+1)/6
Cubes	i^3	[n*(n+1)/2]^2
Arithmetic	a + (i-1)*d	n/2 * (2a + (n-1)*d)
Geometric (r not = 1)	a * r^(i-1)	a * (1 - r^n) / (1 - r)
Harmonic	1/i	approx ln(n) + 0.5772

All results are computed to full JavaScript double precision (about 15-16 significant digits) and displayed with up to 6 decimal places.