Convergence Divergence Test Calculator

Series term a_n

Use n, ^, sqrt(), abs(), log(), sin(), cos(), exp().

Test method

Starting index n

Computed terms

Decision tolerance

p value for p-series

Common ratio r

Student notes

Actions

Example Data Table

Expression	Suggested test	Expected decision	Reason
1/n^2	P-series	Convergent	p = 2 is greater than 1.
1/n	P-series	Divergent	p = 1 gives the harmonic series.
(-1)^(n+1)/n	Alternating	Convergent	Absolute terms decrease toward zero.
(3/4)^n	Geometric	Convergent	The common ratio has absolute value below 1.
n/(n+1)	Nth term	Divergent	Terms approach 1, not zero.

Formula Used

Nth term test: If lim a_n is not 0, then the series diverges.

Ratio test: L = lim |a_(n+1) / a_n|. If L < 1, the series converges. If L > 1, it diverges.

Root test: L = lim |a_n|^(1/n). If L < 1, the series converges. If L > 1, it diverges.

P-series test: The series sum 1/n^p converges when p > 1. It diverges when p ≤ 1.

Geometric test: The series sum ar^n converges when |r| < 1. Its sum is a / (1 - r).

Alternating test: If b_n decreases to 0, then sum (-1)^n b_n converges.

How to Use This Calculator

Enter the term expression using n as the index.
Select auto review or choose one specific convergence test.
Set the starting index and number of computed terms.
Enter p or r when using those special tests.
Press Calculate to show the result above the form.
Use CSV or PDF download for saved work.

About the Convergence Divergence Test Calculator

This calculator helps you study infinite series with organized tests. It accepts a term expression in n. Then it estimates behavior from many computed terms. It is designed for practice, checking homework steps, and building reports. The tool does not replace a written proof. It gives structured evidence that guides the next step.

Why Series Testing Matters

A series can look calm while its sum grows forever. Another series can have large early terms yet still settle. Convergence tests help separate these cases. The nth term test checks a basic requirement. Ratio and root tests work well for powers, factorials, and exponential patterns. The p series and geometric tests handle two common families. Alternating tests review sign changes and decreasing absolute terms.

Advanced Numerical Review

The calculator computes partial sums, tail terms, ratios, and roots. It also checks sign patterns and decreasing tails. You can change the starting index, term count, and tolerance. These controls are useful for slow series. Harmonic style series may need more terms. Fast exponential series usually show a clear pattern quickly. The result explains which test gave the strongest signal.

Best Use Cases

Use the calculator when you know the general term. Try expressions such as 1/n^2, (-1)^(n+1)/n, or (3/4)^n. Choose auto review for a broad scan. Select a single test when your class requires one method. Add a p value for a p series. Add a ratio value for a geometric series. Compare the output with your own algebraic work.

Interpreting Results

A convergent result means the evidence supports a finite sum. A divergent result means the evidence supports no finite sum. An inconclusive result means the selected test cannot decide. This is common when a limit equals one. It also happens with slow decay or mixed behavior. In that case, try another test. You can also raise the term count and review partial sums. Always write the final answer with the theorem used.

Study Tip

Record the expression, chosen test, estimated limit, and final reason. This habit makes your work easier to audit. It also helps you spot mistakes in algebra, index choice, or sign handling before submission. Save exports when comparing several similar series forms later.

FAQs

What does this calculator test?

It reviews infinite series using numeric evidence from ratio, root, nth term, p-series, geometric, and alternating tests.

Can it prove every series result?

No. It supports study and checking. Some series require symbolic comparison, integral work, or a special theorem.

Why can a result be inconclusive?

A test is inconclusive when its limit equals one, data is slow, or the selected theorem does not apply clearly.

How do I enter powers?

Use the caret symbol. For example, enter 1/n^2, (3/4)^n, or (-1)^(n+1)/n.

What starting index should I use?

Use the index from your problem. If none is given, n = 1 is common for most elementary series.

Why add a p value?

The p value lets the p-series rule decide quickly. The series 1/n^p converges only when p is greater than 1.

What does the ratio estimate mean?

It estimates |a_(n+1)/a_n| near the tail. Values below 1 support convergence. Values above 1 support divergence.

Can I export my work?

Yes. Use the CSV button for spreadsheet records. Use the PDF button after a result appears on the page.