Track partial sums with flexible term inputs. Review tables, convergence clues, and instant exports easily. Understand series behavior using clear cumulative results from today.
an = a1 + (n - 1)d
Sn = n / 2 × [2a1 + (n - 1)d]
an = a1rn - 1
Sn = a1(1 - rn) / (1 - r), when r ≠ 1
an = c / n
Sn = c × Σ(1 / k), for k = 1 to n
an = c / np
Sn = c × Σ(1 / kp), for k = 1 to n
Sn is the running total of the values you provide.
Example using an arithmetic sequence with a1 = 2 and d = 3.
| n | Term an | Partial Sum Sn |
|---|---|---|
| 1 | 2 | 2 |
| 2 | 5 | 7 |
| 3 | 8 | 15 |
| 4 | 11 | 26 |
| 5 | 14 | 40 |
A sequence of partial sums shows how a series builds one term at a time. Each new term updates the running total. That running total becomes a new sequence. This idea is central in algebra, calculus, and numerical analysis. It helps students see whether a series appears to settle, grow, or oscillate. It also helps teachers present abstract notation through concrete values and tables. Partial sums also support estimation. A finite partial sum can approximate an infinite total when convergence is present. That makes the topic practical.
Partial sums reveal useful patterns quickly. Arithmetic terms usually push totals upward or downward steadily. Geometric terms can explode, shrink, or alternate. Harmonic growth is slower, yet it still keeps increasing. A p-series can behave very differently depending on the exponent. When you inspect several partial sums together, you can spot stability, divergence, or slow convergence before formal proof. The chart in this calculator makes those patterns easier to notice. Visual evidence is not a proof, yet it guides better checking today.
This calculator supports arithmetic, geometric, harmonic, p-series, and custom inputs. You can enter values, choose a model, and set the number of terms. The tool then lists each term beside its matching cumulative sum. That structure is useful for homework checks, tutoring sessions, and classroom demonstrations. It is also helpful for self-study. You can compare models, test conjectures, and build intuition with less manual work. The result summary highlights the last term, the final finite sum, and a convergence note. This saves time during revision and reduces arithmetic mistakes.
The export options make review easier. Save the table as CSV for spreadsheet work. Save it as PDF for notes or assignments. The example table shows how partial sums grow across several terms. The formula section explains the rule behind each supported sequence type. The how-to section keeps the workflow simple. Enter the inputs, calculate, read the table, and inspect the graph. This process builds confidence with sequences, series, convergence tests, and cumulative reasoning in everyday study. It also supports exam preparation, lesson planning, and answer verification.
A partial sum is the running total of the first n terms of a series. If the terms are a1, a2, a3, the third partial sum is a1 + a2 + a3.
They help you study convergence, estimate infinite sums, and understand how a series behaves term by term. They also make abstract series notation easier to visualize.
It gives a convergence hint based on the selected sequence model. That hint is useful for learning, but formal convergence proofs still require mathematical reasoning.
A series is the expression formed by adding terms. Partial sums are the finite running totals created from the first 1, 2, 3, or n terms.
No. An infinite geometric series converges only when the absolute value of the ratio is less than 1. Otherwise, the partial sums do not settle to a finite limit.
Its terms become small, but not fast enough. The partial sums keep increasing without approaching a fixed finite value, so the infinite harmonic series diverges.
Yes. Use the custom list option and enter numbers separated by commas, spaces, or new lines. The calculator will build partial sums from the provided values.
They export the result table containing the term index, term value, and partial sum. This is useful for reports, homework review, and spreadsheet analysis.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.