Calculator Inputs
Example Data Table
| Series Type | Inputs | Converges? | Infinite Sum |
|---|---|---|---|
| Geometric | a = 12, r = 0.5, start n = 1 | Yes | 24 |
| Geometric | a = 8, r = 1.2, start n = 1 | No | Diverges |
| p-Series | c = 1, p = 2, start n = 1 | Yes | ≈ 1.644934 |
| Telescoping | a = 1, k = 2, start n = 1 | Yes | 0.75 |
Formula Used
1) Geometric series
For Σ a·r^(n-start), the infinite sum exists only when |r| < 1. Then:
S = a / (1 - r)
2) p-series
For Σ c / n^p, the infinite sum exists only when p > 1. This page estimates the total numerically using many terms plus a tail correction.
3) Telescoping series
For Σ a / [n(n+k)], partial fractions give:
a / [n(n+k)] = (a/k)·(1/n - 1/(n+k))
Most terms cancel, leaving a short exact expression for the infinite sum.
How to Use This Calculator
- Select the series family you want to evaluate.
- Enter the start index and preview term count.
- Fill the fields for that chosen series type.
- Press Calculate Infinite Sum.
- Read the convergence decision, infinite sum, and preview partial sum.
- Use the Plotly graph to inspect term decay and partial sum behavior.
- Download your result summary as CSV or PDF when needed.
Frequently Asked Questions
1. What is an infinite series?
An infinite series adds endlessly many terms from a sequence. We study whether its partial sums settle toward one fixed value. If they do, the series converges and that limiting value is its sum.
2. Why does convergence matter?
Convergence decides whether an infinite sum has a meaningful finite answer. Without convergence, adding more terms will not stabilize near one number, so reporting a final sum would be misleading.
3. When does a geometric series converge?
A geometric series converges exactly when the common ratio satisfies |r| < 1. Then each term shrinks enough for the partial sums to approach a fixed limit.
4. What is a p-series?
A p-series has the form Σ 1/n^p or a scaled version of it. It converges when p > 1 and diverges when p ≤ 1.
5. What makes a telescoping series special?
Telescoping series simplify after algebraic splitting. Many middle terms cancel between consecutive partial sums, so the infinite total can often be written exactly from only a few surviving terms.
6. Why is the p-series result approximate here?
Most p-series totals do not simplify into an elementary closed form. This calculator adds many terms and estimates the remaining tail, which gives a strong practical approximation for convergent cases.
7. What does preview partial sum mean?
It is the sum of the first chosen number of displayed terms. Comparing it with the infinite sum shows how quickly the series is approaching its limit.
8. Can this calculator prove divergence?
It can confirm divergence for the included rule types using known tests. For example, geometric series fail when |r| ≥ 1, and p-series fail when p ≤ 1.