Enter your series type (arithmetic, geometric, or infinite geometric), along with the first term, common difference or ratio, and number of terms — and the Sum of Series Calculator returns the partial sum or infinite sum, plus a term-by-term breakdown. Works for both convergent infinite series and finite partial sums.
Sum of Series
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Last Term (nth Term)
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A series is the sum of the terms of a sequence. For example, if your sequence is 1, 2, 3, 4, 5, the corresponding series is 1 + 2 + 3 + 4 + 5 = 15. Series can be finite (a set number of terms) or infinite (continuing without end).
For an arithmetic series with first term a, common difference d, and n terms, use the formula: Sₙ = (n/2) × [2a + (n − 1)d]. Alternatively, Sₙ = (n/2) × (first term + last term). This works because pairs of terms equidistant from each end always add up to the same value.
For a finite geometric series with first term a, common ratio r, and n terms, use: Sₙ = a × (1 − rⁿ) / (1 − r), when r ≠ 1. If r = 1, all terms are equal and Sₙ = a × n. The ratio r is found by dividing any term by the one before it.
An infinite geometric series converges (has a finite sum) only when the absolute value of the common ratio |r| < 1. The formula is S = a / (1 − r), where a is the first term. If |r| ≥ 1, the series diverges and has no finite sum.
In an arithmetic series, consecutive terms differ by a constant value called the common difference (d), e.g. 2, 5, 8, 11 (d = 3). In a geometric series, consecutive terms share a constant ratio (r), e.g. 3, 6, 12, 24 (r = 2). The type of series determines which sum formula to apply.
Sigma notation is a compact way to write the sum of a series. The symbol Σ (Greek capital letter sigma) means 'sum of'. For example, Σ(n=1 to 5) n means 1 + 2 + 3 + 4 + 5. The expression below Σ is the starting index, above it is the ending index, and to the right is the general term formula.
The sum of all integers from 1 to N is given by S = N × (N + 1) / 2. This is a special case of an arithmetic series with a = 1 and d = 1. For example, the sum of 1 to 100 is 100 × 101 / 2 = 5050, famously computed by the mathematician Gauss.
For geometric series, the key test is whether |r| < 1 (converges) or |r| ≥ 1 (diverges). A necessary condition for any series to converge is that the individual terms must approach zero. More advanced tests like the ratio test, root test, or integral test are used for non-geometric series.