Sum of Series Calculator - Arithmetic, Geometric & More
Calculate the sum of arithmetic sequences, geometric series, harmonic series, and sum-of-squares series instantly with step-by-step formulas.
Select a series type, enter the required parameters, and get the exact sum with the formula applied.
Sum of Series Calculator - Arithmetic, Geometric & More
Calculate the sum of arithmetic sequences, geometric series, harmonic series, and sum-of-squares series instantly with step-by-step formulas.
About the sum of series calculator
A mathematical series is the sum of the terms of a sequence. While a sequence is simply an ordered list of numbers, a series is what you get when you add those numbers together. The sum of series calculator handles four of the most important and widely-encountered series in mathematics: arithmetic sequences, geometric series, harmonic series, and the sum of squares of the first n natural numbers.
An arithmetic sequence is one in which consecutive terms differ by a fixed amount called the common difference. The sequence 1, 3, 5, 7, 9 has a first term of 1 and a common difference of 2. The sum of the first n terms of an arithmetic sequence is given by the formula S = (n/2)(2a + (n−1)d), where a is the first term and d is the common difference. This formula is attributed to Gauss, who reportedly derived it as a schoolboy when his teacher asked the class to sum the integers from 1 to 100. Gauss noticed that pairing the first and last terms always gave 101, and there were 50 such pairs, yielding a sum of 5050.
A geometric series is one in which each term is obtained by multiplying the previous term by a fixed ratio r. The sequence 2, 4, 8, 16 has a first term of 2 and a common ratio of 2. The sum of the first n terms of a geometric series is S = a(1 − rⁿ)/(1 − r) when r ≠ 1. When r = 1, all terms are equal and the sum is simply n × a. Geometric series are fundamental in finance — compound interest, annuities, and loan repayments all involve geometric sums.
The harmonic series is defined as H(n) = 1 + 1/2 + 1/3 + 1/4 + … + 1/n. Unlike arithmetic and geometric series, the harmonic series does not have a simple closed-form expression for a partial sum, so the calculator computes it directly by adding the reciprocals of the integers from 1 to n. The harmonic series is famous for being divergent — it grows without bound as n increases — although it does so extremely slowly. H(10) ≈ 2.93, H(100) ≈ 5.19, and H(1,000,000) ≈ 14.39.
The sum of squares formula gives the total of 1² + 2² + 3² + … + n², which equals n(n+1)(2n+1)/6. This elegant closed form can be proved by mathematical induction and arises naturally in statistics (variance and standard deviation), numerical methods, and combinatorics. For example, the sum of squares of the first 15 natural numbers is 15 × 16 × 31 / 6 = 1240.
Understanding which series type applies to your problem is the key first step. If consecutive terms differ by a constant, use arithmetic. If consecutive terms are related by a constant multiplier, use geometric. If you are summing reciprocals of integers, use harmonic. If you are summing squared integers, use sum of squares. The calculator guides you through the appropriate input fields for each type and displays the formula used so you can verify the calculation independently.
Sum of series examples
Four worked examples illustrating each series type with realistic inputs and verified results.
| Series | Sum | Formula applied |
|---|---|---|
| 1 + 3 + 5 + … (first 10 odd numbers, a=1, d=2, n=10) | 100 | Arithmetic sequence. S = (10/2)(2×1 + 9×2) = 5 × 20 = 100. |
| 2 + 4 + 8 + … (powers of 2, a=2, r=2, n=8) | 510 | Geometric series. S = 2(1 − 2⁸)/(1 − 2) = 2 × 255 = 510. |
| 1 + 1/2 + 1/3 + … (harmonic series, n=20) | ≈ 3.5977 | Harmonic partial sum. Computed by direct addition of reciprocals 1 through 20. |
| 1² + 2² + 3² + … + 15² (sum of squares, n=15) | 1240 | Sum of squares formula. S = 15 × 16 × 31 / 6 = 1240. |
How to use the sum of series calculator
- Select the series type that matches your problem: Arithmetic Sequence, Geometric Series, Harmonic Series, or Sum of Squares.
- Enter the required parameters. Arithmetic needs First Term (a), Common Difference (d), and Number of Terms (n). Geometric needs First Term (a), Common Ratio (r), and n. Harmonic and Sum of Squares need only n.
- Click Calculate Sum. The result appears immediately along with the formula used.
- Use the example buttons below the table to instantly load a worked example for any series type.
- Click Reset to clear all fields and start a fresh calculation.
Sum of series FAQ
What is the difference between a sequence and a series?
A sequence is an ordered list of numbers, such as 1, 3, 5, 7. A series is the sum of all the terms in a sequence: 1 + 3 + 5 + 7 = 16. The sum of series calculator computes the series (the total), not the individual terms of the sequence.
How do I find the sum of an arithmetic sequence?
Use the formula S = (n/2)(2a + (n−1)d), where a is the first term, d is the common difference, and n is the number of terms. Alternatively, S = n × (first term + last term) / 2, which is Gauss's pairing shortcut. Enter a, d, and n in the calculator to get the result instantly.
When does the geometric series formula not apply?
The standard formula S = a(1 − rⁿ)/(1 − r) cannot be used when the common ratio r equals 1, because it would produce a division by zero. When r = 1, every term equals the first term a, so the sum is simply n × a. The calculator handles this case automatically.
Does the harmonic series converge?
No. The harmonic series 1 + 1/2 + 1/3 + … diverges — it grows without limit as n increases, albeit very slowly. The calculator computes partial sums (the sum of a finite number of terms), which are always finite. The divergence means there is no single finite value the full series approaches.
What is the sum-of-squares formula used for?
The formula n(n+1)(2n+1)/6 appears in statistics (calculating variance from raw scores), in numerical integration methods, and in combinatorics problems. It gives the exact total of 1² + 2² + … + n² for any positive integer n without needing to square and add each term individually.
Can I use this calculator for very large n values?
Yes, for arithmetic, geometric, and sum-of-squares series, the calculator uses closed-form formulas that are accurate for any n. For the harmonic series, the calculator sums 1/1 through 1/n directly, so very large n (e.g., n = 100,000) may take a brief moment. All results are computed in double-precision floating-point arithmetic accurate to about 15 significant digits.