Series Convergence Tests Calculator

Calculator

Series term model

Preferred test

Coefficient a

Common ratio r

Power p

Log power q

Exponential base b

Starting index n0

Upper sample index

Tail sample window

Zero-term tolerance

Output precision

Custom term a(n) Use n, +, -, *, /, ^, pow(), sqrt(), log(), sin(), cos(), exp(), abs().

Example Data Table

Series	Term Model	Key Input	Expected Result
1 / n²	p-series	p = 2	Convergent
1 / n	p-series	p = 1	Divergent
(1/2)ⁿ	Geometric	r = 0.5	Convergent
(-1)ⁿ / n	Alternating p-series	p = 1	Conditionally convergent

Formula Used

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

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

Logarithmic p-series: Σ 1 / (n^p(ln n)^q) converges when p > 1, or when p = 1 and q > 1.

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

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

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

How To Use This Calculator

Select the term model that matches your series.
Enter coefficient, ratio, powers, base, or a custom expression.
Choose the starting index and upper sample index.
Pick a preferred test, or keep auto select.
Press Calculate to show the result above the form.
Use CSV or PDF buttons to save the output.

Understanding Series Convergence

A series adds infinitely many terms. The main question is simple. Does the running total approach one fixed number, or does it drift forever? This calculator helps study that question with common tests. It does not replace proof. It gives structured evidence and clean intermediate values.

What The Calculator Checks

The tool supports geometric, p-series, logarithmic p-series, alternating p-series, factorial ratio, and custom numeric terms. Each model has different strengths. Geometric and p-series cases can often be judged directly. Custom expressions need numeric sampling, so the result may be marked inconclusive.

Why Multiple Tests Matter

One convergence test rarely solves every problem. The ratio test is strong for factorials and powers. The root test works well when the nth power is visible. The p-series comparison is useful when a term behaves like one divided by a power of n. The nth-term test quickly catches series whose terms do not approach zero.

Reading The Result

A convergent result means the series appears to approach a finite sum. Absolute convergence means the positive version also converges. Conditional convergence usually appears in alternating series. Divergence means the series cannot settle to a finite total. Inconclusive means the selected evidence was not strong enough.

Numerical Limits

Computers only check finitely many terms. A slow series can fool numeric sampling. The harmonic series is a classic example. Its terms go to zero, but its sum still diverges. Always compare the output with a known theorem when precision matters.

Good Input Practice

Start with a clear formula. Use a larger upper index for slow terms. Keep the tolerance small for sensitive tests. For custom expressions, write the term as a function of n. Avoid formulas with undefined values near the starting index.

Common Uses

Students can test homework examples before writing proofs. Teachers can prepare answer checks. Analysts can inspect tail behavior in probability, estimation, and approximation work. The export buttons make it easy to keep a record of assumptions, sample size, and conclusion.

Final Advice

Use the calculator as a guide. Then write the mathematical reason. It also encourages careful reasoning before formal work begins in class or research. A clear test, valid conditions, and a final conclusion make the solution complete.

FAQs

What is a series convergence test?

It is a method used to decide whether an infinite sum approaches a finite value. Common tests include p-series, geometric, ratio, root, comparison, and alternating tests.

Can this calculator prove convergence?

It gives exact decisions for supported standard models. Custom expressions use numeric evidence. For formal work, always write the theorem and verify its conditions.

What does absolute convergence mean?

Absolute convergence means the series still converges after every term is changed to its positive magnitude. It is stronger than conditional convergence.

What does conditional convergence mean?

Conditional convergence means the original series converges, but the absolute value series diverges. Alternating harmonic style series often behave this way.

Why is my custom result inconclusive?

Some tests have limit values near one. Numeric samples cannot always decide these cases. Increase the sample index or use a more specific model.

Which test should I choose?

Use geometric for powers of a constant. Use p-series for 1/n powers. Use ratio for factorials. Use alternating for sign-changing decreasing terms.

Why does the nth-term test matter?

If terms do not approach zero, the series must diverge. This test is fast and often catches impossible convergence immediately.

Can I export my answer?

Yes. Use the CSV button for spreadsheet work. Use the PDF button for a simple printable record of the result and key estimates.