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.

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

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