Convergent or Divergent Series Calculator

Series Calculator

General term a_n

Use n, +, -, *, /, ^, sqrt, log, sin, cos, exp, or pow.

Test method

Known pattern

Starting index n

Terms to compute

Rows to display

p value

First geometric term a

Common ratio r

Comparison p

Tolerance

Decimal precision

Formula Used

Nth-term test: If lim a_n is not zero, then the series diverges.

Ratio test: L = lim |a_n+1 / a_n|. The series converges when L < 1 and diverges when L > 1.

Root test: L = lim sup |a_n|^1/n. The series converges when L < 1 and diverges when L > 1.

p-series: The series Σ 1/n^p converges when p > 1 and diverges when p ≤ 1.

Geometric series: The series Σ arⁿ converges when |r| < 1. Its sum is a / (1 - r).

Alternating test: An alternating series converges when term magnitudes decrease to zero.

Comparison test: Compare the given tail with a known benchmark, such as 1/n^p.

How to Use This Calculator

Enter the general term using n as the index.
Choose auto mode or select a specific convergence test.
Enter p-series, geometric, or comparison values when needed.
Set the starting index, term count, tolerance, and precision.
Press the submit button to view results above the form.
Download the result as CSV or PDF for later study.

Example Data Table

Series Term	Suggested Test	Expected Result	Reason
`1/n^2`	p-series	Convergent	p = 2, and p is greater than 1.
`1/n`	p-series	Divergent	p = 1 creates the harmonic series.
`(1/3)^(n-1)`	Geometric	Convergent	The common ratio has absolute value below 1.
`(-1)^(n+1)/n`	Alternating	Convergent	Magnitudes decrease toward zero.
`n/(n+1)`	Nth-term	Divergent	The term does not approach zero.

About This Series Decision Tool

Infinite series appear in calculus, analysis, physics, finance, and engineering models. A series adds infinitely many terms. The main question is simple. Does the sum settle to a finite value, or does it fail to settle? This calculator gives a structured numerical answer. It also explains which test produced the strongest evidence.

What the Calculator Checks

The tool can inspect a general term, such as 1/n^2, (-1)^n/n, or n/2^n. It estimates partial sums. It reviews the last computed term. It also applies ratio and root tests when the term values allow those comparisons. You may choose a direct p-series or geometric test when the pattern is known. These direct tests are stronger because they use exact rules.

Why Multiple Tests Help

No single convergence test works for every series. The nth-term test can prove divergence, but it cannot prove convergence. The ratio test works well for factorials and exponential terms. The root test is useful when powers of n appear inside the term. Alternating signs need a different check. A decreasing positive magnitude that approaches zero may pass the alternating series test.

Reading the Result

The verdict shows convergent, divergent, likely convergent, or inconclusive. A likely label means the numerical evidence is helpful, but it is not a proof. The table shows early terms and partial sums. These values reveal growth, cancellation, and settling behavior. Use more terms and tighter tolerance for slow series, such as harmonic-like inputs.

Best Use Cases

Use this page while studying calculus or checking homework steps. It is also useful when building examples for lessons, notes, or quick reports. The export buttons save your computed evidence for later review. Always confirm important results with a formal written test, especially when the calculator says inconclusive.

Accuracy Tips

Enter the term using n as the index. Keep parentheses clear around denominators, exponents, and alternating factors. A larger maximum term count improves numerical checks, but it may take longer. A very small tolerance can expose slow convergence. If the expression has singular values near the starting index, move the start value forward. When a known pattern exists, choose that test first, because exact identities beat numerical guesses. Record assumptions with each export.

FAQs

Can this calculator prove every convergence result?

No. Direct p-series and geometric checks follow exact rules. Other modes use numerical evidence. Use the result as guidance, then support important answers with a written theorem-based proof.

What should I enter for the term?

Enter the formula for a single term using n. Examples include 1/n^2, (-1)^n/n, sqrt(n)/(n^2+1), and (1/3)^(n-1).

Why does the answer say inconclusive?

Some tests fail when their limit equals 1 or when the numerical tail is unclear. Try another test, increase term count, or use a known comparison.

Which test should I choose first?

Use p-series or geometric mode when the pattern is clear. Use ratio for factorial or exponential terms. Use root for terms with large powers.

Does a small last term prove convergence?

No. A zero term limit is required, but it is not enough. The harmonic series has terms approaching zero, yet it diverges.

What does likely convergent mean?

It means the computed evidence supports convergence. It may not be a formal proof. Slow or unusual series can need deeper analysis.

Can I export the result?

Yes. Use the CSV button for spreadsheet data. Use the PDF button for a simple report with the main verdict and computed table.

Why should I increase maximum terms?

More terms can reveal slow convergence, delayed divergence, or partial sum behavior. It also improves ratio, root, and tail estimates.