Calculate Sum of Series Online

Calculator Inputs

Series type

Number of terms

Start index

First term, a

Common difference, d

Common ratio, r

Coefficient, c

Power exponent, p

Add constant to each term

Multiply each term by

Decimal precision

Visible table rows

Custom term expression

Allowed items include n, pi, e, +, -, *, /, ^, parentheses, sqrt, abs, sin, cos, tan, log, log10, exp, floor, ceil, and pow.

Alternate signs after the first term

Example Data Table

Series Type	Inputs	First Terms	Expected Sum
Arithmetic	a = 2, d = 3, terms = 5	2, 5, 8, 11, 14	40
Geometric	a = 3, r = 2, terms = 4	3, 6, 12, 24	45
Harmonic	c = 1, start = 1, terms = 4	1, 1/2, 1/3, 1/4	2.083333
Power	c = 1, p = 2, start = 1, terms = 4	1, 4, 9, 16	30
Custom	f(n) = n^2 + 1, start = 1, terms = 3	2, 5, 10	17

Formula Used

The calculator uses finite partial sums. It builds each term, then adds it to a running total.

Arithmetic: S_n = n / 2 × [2a + (n - 1)d].

Geometric: S_n = a(1 - rⁿ) / (1 - r). When r = 1, S_n = an.

Harmonic: S_n = c × Σ(1 / k), using your selected starting index.

Power: S_n = c × Σ(k^p), using your selected starting index.

Custom: S_n = Σf(k), where the expression uses n as the index.

How to Use This Calculator

Select the series type that matches your pattern.
Enter the number of terms and starting index.
Fill the fields needed by that series type.
Use the custom expression box when no preset fits.
Set precision and visible table rows.
Press Calculate to show the answer below the header.
Use CSV or PDF buttons when you need an export.

About This Sum of Series Calculator

A series is the total of ordered terms. Each term follows a rule. This calculator helps you test that rule fast. It supports arithmetic, geometric, harmonic, power, and custom forms. You can change the first term, ratio, difference, exponent, index, precision, and number of terms.

Why Series Sums Matter

Series appear in budgets, schedules, physics, coding, games, loans, and classroom work. A small pattern can grow quickly. A clear sum can prevent mistakes. It can also show how a sequence behaves as more terms are added.

Advanced Inputs

The arithmetic option uses a first term and common difference. The geometric option uses a first term and common ratio. The harmonic option adds reciprocal style terms. The power option sums indexed powers. The custom option accepts an expression using n. This makes the tool flexible for many simple models.

Reading the Result

After you submit the form, the result appears above the form. You can review the sum, first term, last term, average term, smallest term, largest term, and formula note. A term table shows each visible index. It also shows the running total.

Useful Exports

Use the CSV export when you need spreadsheet work. Use the PDF export when you need a shareable summary. Both exports use the same inputs. This keeps records consistent.

Good Practice

Choose a sensible number of terms. Very large counts may be slow. Check the first few terms before trusting the final sum. For a custom expression, use n for the index. Use parentheses when an expression may be unclear. Compare the table with your expected pattern. This habit catches most entry errors.

Limits and Accuracy

The calculator uses decimal arithmetic for live pages and exports. Results are rounded only for display. Internally, the loop keeps full available precision. Arithmetic and geometric formulas also have direct notes, so you can compare the loop with the known rule. Custom expressions should stay within ordinary numeric ranges. Division by zero, invalid syntax, and missing values are blocked when possible. If a series converges only in theory, this page still reports the finite partial sum you requested. That makes the answer practical, honest, and easy to reproduce during later review checks too.

FAQs

What is a sum of series?

It is the total found by adding terms in a sequence. The terms may follow arithmetic, geometric, harmonic, power, or custom rules.

Can this calculator handle geometric series?

Yes. Enter the first term, common ratio, and number of terms. The calculator shows the finite sum and a convergence note.

How do I enter a custom series?

Select Custom expression. Type a term rule using n as the index. For example, use n^2 + 1 or 1/n.

Does the tool show each term?

Yes. It creates a term table with the term number, index, term value, and running total. You can limit visible rows.

What does the start index mean?

It is the first index used for harmonic, power, and custom calculations. Arithmetic and geometric presets count from the first entered term.

Can I export the result?

Yes. Use the CSV button for spreadsheet data. Use the PDF button for a compact summary and visible term list.

Why does my harmonic series give an error?

A harmonic term cannot divide by zero. Change the start index if it creates index 0 during the requested term range.

Is the answer an infinite series sum?

No. The result is a finite partial sum based on your term count. Convergence notes only explain the related infinite behavior.