Partial Sum Calculator
Compute finite series totals from a general term across an index range, showing steps, formula preview, and clear numerical results.
Inputs
Equation Preview
Helping Notes
Use a single index like n or r. Variables are autodetected from your expression when you omit an index name.
Functions like sin, cos, exp, log, factorial(n), and binomial(n,k) are supported. Enter integers for start and end.
Results
Inputs Summary
Partial Sum
Computation Details
What Is a Partial Sum Calculator?
A Partial Sum Calculator finds the sum of the first terms of a sequence, or more generally the sum from index to . Given a term rule (explicit or generated), it builds the correct closed‑form when available—such as arithmetic or geometric series—or evaluates numerically with exact fractions/decimals. The tool displays sigma notation, substitutes bounds, and simplifies to a clean result, making it easy to check homework, engineering totals, or finance accumulations without manual algebra. Where appropriate, it also shows telescoping behavior, convergence notes, and how partial sums approach a limit for convergent infinite series.
About the Partial Sum Calculator
The core definition is . For arithmetic sequences with first term and common difference , the partial sum is . For geometric sequences with ratio , (equivalently ). Partial sums over a sub‑range use the difference rule . Telescoping sums exploit cancellations, e.g., . The calculator recognizes these templates automatically, and when a closed form is not standard, it computes exact finite sums directly from the term definition.
Definition:
Arithmetic: or
Geometric (r≠1):
Sub‑range m..n:
Telescoping example:
Index shift:
How to Use This Partial Sum Calculator
- Choose sequence type or enter a general term . Set bounds (e.g., or ).
- If arithmetic, provide and . If geometric, provide and .
- Press calculate. The tool shows sigma notation, substitutions, simplification steps, and the final partial sum.
- Copy the result (exact form and optional decimal) and, if needed, view a convergence note for the infinite‑series limit.
Examples
- Arithmetic: ⇒ .
- Geometric: ⇒ .
- Sub‑range (arithmetic): ⇒ .
- Telescoping: ⇒ for , .
Formula Snippets Ready for Rendering
FAQs
What is a partial sum?
It’s the sum of the first n terms of a sequence, or the sum over any finite index range.
How is a partial sum different from an infinite series sum?
A partial sum is finite. An infinite series sum, if it exists, is the limit of partial sums as n → ∞.
Do I need a closed form for ak?
No. The calculator can sum finite terms numerically; closed forms are used when available (e.g., arithmetic, geometric).
Can I start at k = 0 instead of 1?
Yes. The bounds are flexible; formulas adapt via index shifts to accommodate your convention.
What happens when r = 1 in a geometric series?
Then Sn = n·a1 because every term equals a1; the standard fraction formula would divide by zero.
How do I sum from m to n?
Compute Sn − Sm−1. The tool automates this and shows the substitution clearly.
What is a telescoping sum?
A sum where intermediate terms cancel after rewriting, leaving only a few terms—very fast partial sums.
Will rounding affect results?
Exact arithmetic is retained by default; optional decimals are provided for quick numeric checks.
Can I mix fractions and decimals?
Yes, inputs are normalized internally; outputs can be exact fractions or decimals per your preference.
Does the calculator handle alternating signs?
Yes. It sums terms with (−1)k patterns exactly or numerically and shows the resulting closed form when available.