Find the Limit of the Sequence Calculator

Calculator

Sequence formula a_n

Use n as the variable. Example: (0.8)^n, (n^2+1)/(n^2-4), (-1)^n

Start n

End n

Step

Analysis points

Comma-separated large n values used for estimating the limit.

Convergence tolerance

Infinity threshold

Display decimals

Reset

Supported functions: sin, cos, tan, asin, acos, atan, sqrt, abs, exp, log, ln, log10

Tip: explicit multiplication is safest, such as 2*n instead of 2n.

Example Data Table

Sample sequence: a_n = (2n + 1) / (n + 3)

n	a_n	Observation
1	0.750000	Early term is far from the final trend.
5	1.375000	Values start rising toward a stable number.
10	1.615385	The sequence is still below its limit.
50	1.943396	Large n pushes the value closer to 2.
100	1.970874	The limit estimate is clearly approaching 2.

Formula Used

Core numeric idea:

The calculator evaluates a_n for increasingly large values of n and compares later terms to detect convergence, divergence, oscillation, or growth toward infinity.

Finite limit test: if later terms become very close, the calculator estimates

L ≈ average of the last few large-n terms

Zero limit pattern: if |a_n| keeps shrinking and approaches 0, the limit is estimated as 0.

Positive or negative infinity: if terms keep growing in one direction and cross the infinity threshold, the result is treated as +∞ or -∞.

Oscillation detection: if the sign keeps flipping and the values do not settle, the calculator marks the sequence as divergent.

Useful theory examples:

For a_n = 1 / n^p with p > 0, the limit is 0.
For a_n = rⁿ, the limit is 0 when |r| < 1.
For equal-degree rational sequences, the limit is the ratio of leading coefficients.

How to Use This Calculator

Enter the sequence formula using n as the variable.
Set the graph range with start n, end n, and step.
Add large analysis points such as 100, 500, and 1000.
Adjust tolerance if you want stricter convergence checks.
Click Find Limit to generate the estimate, table, and graph.
Download the values as CSV or save the result summary as PDF.

Frequently Asked Questions

1) What does the calculator actually estimate?

It estimates the long-term behavior of a sequence by checking values at large n. It can suggest a finite limit, zero, positive infinity, negative infinity, or no limit.

2) Does this give a formal proof?

No. It is a numeric estimator. It is very useful for checking patterns quickly, but exam or textbook proofs still need algebraic or theoretical limit arguments.

3) Can I enter rational sequences?

Yes. Expressions like (3*n^2+1)/(2*n^2-5) work well. Rational sequences are often among the easiest to estimate because their large-n behavior becomes clear quickly.

4) Can it detect oscillation?

Yes. Sequences such as (-1)^n often alternate without settling. The calculator checks repeated sign changes and reports that the limit does not exist.

5) Why should I use large analysis points?

Large values of n reveal the long-run trend better. Small terms can be misleading, especially when the sequence converges slowly or changes direction early.

6) What formulas are supported?

You can use arithmetic operators, powers, parentheses, constants like pi and e, and functions such as sin, cos, tan, sqrt, abs, exp, log, ln, and log10.

7) What if some terms are undefined?

The calculator skips undefined sampled terms and reports them. If too many important terms are undefined, the result becomes insufficient or less reliable.

8) When should I change the tolerance?

Use a smaller tolerance when you want a stricter convergence test. Use a slightly larger one if your sequence converges slowly and the default setting feels too strict.