Summation Calculator

How sigma notation works

Sigma notation expresses a sum concisely: the capital Greek letter sigma (Σ) followed by a formula, a variable, and lower and upper bounds. The calculator evaluates the formula at each integer value of the variable from the lower bound to the upper bound and sums the results.

For example, Σ(n, 1, 5) of n^2 means 1^2 + 2^2 + 3^2 + 4^2 + 5^2 = 55. This added context connects the displayed result to the assumptions, method, and practical interpretation shown elsewhere on the page.

The lower bound tells you where the index starts. The upper bound tells you where it stops. The expression to the right of Σ defines the value of each term. Changing any of those three pieces changes the final sum.

Supported expressions

The calculator supports basic arithmetic (+, -, *, /), exponents (^), parentheses, sqrt(), abs(), sin(), cos(), tan(), log(), ln(), and the constants pi and e. The variable defaults to n but can be renamed.

That means you can use the page for polynomial sums, reciprocal sums, shifted formulas, and many finite trig or logarithmic expressions as long as every evaluated term remains finite across the chosen range.

Partial sums versus the final summation

A partial sum is the running total after a finite number of terms. When people search for a sigma notation calculator with steps, they usually want to see not only the final answer but also how each new term changes the total. That is why a running-total preview matters.

If you sum n from 1 to 4, the partial sums are 1, then 3, then 6, then 10. The final summation is 10, but the intermediate totals help you verify that the formula and bounds were entered correctly.

Worked examples you can check quickly

Start with a simple arithmetic series. If the expression is n and the bounds are 1 to 10, the result is 55. This is the classic sum of the first ten natural numbers.

Next try a square series. If the expression is n^2 and the bounds are 1 to 4, the calculator evaluates 1 + 4 + 9 + 16 and returns 30. That is a quick way to verify that exponents are being interpreted correctly.

Shifted expressions are also common in homework and exam work. If the expression is 2*n+1 and the bounds are 0 to 3, the terms are 1, 3, 5, and 7, so the final summation is 16.

Closed-form checks for common finite sums

A summation calculator is useful even when a shortcut formula exists because it lets you confirm the answer term by term. For the sum of n from 1 to N, the closed form is N(N+1)/2. For the sum of n^2 from 1 to N, the closed form is N(N+1)(2N+1)/6.

These identities are useful as verification tools. If your direct calculation and the closed form disagree, the usual cause is an indexing mistake, a shifted formula, or an upper bound entered one step too high or too low.

Why finite-term checks matter

This page is a finite summation calculator, not an infinite-series convergence checker. That distinction matters because a finite sum only needs the calculator to evaluate the chosen terms, while an infinite series requires additional analysis about convergence or divergence.

In practice, the biggest input errors are reversed bounds, non-integer bounds, and formulas that become undefined inside the range, such as 1/(n-2) when n reaches 2. A good sigma notation solver should flag those cases instead of returning a misleading total.