Can a numerical calculator prove a sequence converges?

No. Evaluating finitely many terms can suggest convergence or divergence, but a mathematical proof requires an appropriate argument.

Which functions can I enter?

The calculator supports n, constants pi and e, arithmetic operators, powers, parentheses, and sqrt, abs, exp, ln, log, sin, cos, and tan functions.

Sequence Limit Calculator

Estimate the limit of a mathematical sequence as n approaches infinity. Enter a formula in n, evaluate its terms at increasingly large values, and inspect the results to explore whether the sequence appears to converge or diverge.

Calculate a Sequence Limit

Sequence expression aₙ

Enter an expression using n, arithmetic operators, parentheses, and supported functions.

Starting n

The calculator evaluates terms at progressively larger n.

Supported syntax: n, pi, e, +, -, *, /, ^, parentheses, sqrt(x), abs(x), exp(x), ln(x), log(x), sin(x), cos(x), and tan(x). Use explicit multiplication, such as 2*n instead of 2n.

Reset

Try an example

What Is the Limit of a Sequence?

A sequence is an ordered list of values written as a₁, a₂, a₃, and so on. Its limit describes the value that the terms approach as the integer index n increases without bound.

Notation: lim_n→∞ aₙ = L

Informally, a sequence converges to L if its terms can be made arbitrarily close to L by choosing n sufficiently large. Not every sequence has a finite limit: some grow without bound, while others oscillate.

Common sequence limits

Reciprocal sequence:
lim_n→∞ 1/n = 0

Ratio of equal-degree polynomials:
lim_n→∞ (2n² + 3n)/(n² + 1) = 2

Exponential limit:
lim_n→∞ (1 + 1/n)ⁿ = e

Oscillating sequence:
aₙ = sin(n) does not approach a single limit as n grows.

How to Determine a Sequence Limit

1. Simplify the expression

Algebraic simplification can reveal the dominant terms and make the limit easier to evaluate. For a rational expression, divide the numerator and denominator by the highest power of n when appropriate.

2. Compare the highest powers

For a ratio of polynomials, the highest-degree terms usually determine the limit. If the numerator and denominator have the same degree, the ratio of their leading coefficients is the limit.

3. Check convergence carefully

A list of numerical terms is useful for intuition, but it cannot prove convergence. A proof may use the definition of a limit, comparison arguments, monotonicity and boundedness, or a known limit theorem.

Frequently Asked Questions

What does n approaching infinity mean?

It means considering sequence terms for larger and larger positive integer values of n. Infinity is not itself a numerical input to the formula.

Can this calculator prove that a sequence converges?

No. It evaluates a finite set of terms and provides a numerical indication. A rigorous proof requires mathematical reasoning about the sequence for all sufficiently large n.

Why might the estimate be inaccurate?

Some sequences converge very slowly, oscillate, or grow beyond the range of ordinary floating-point arithmetic. A finite numerical sample may not capture their long-term behavior.

Can I use square roots and trigonometric functions?

Yes. The calculator supports sqrt, sin, cos, tan, exp, ln, log, and abs, along with arithmetic operations and powers. Use parentheses for function arguments, such as sqrt(n^2+1).

Why must I write 2*n instead of 2n?

The expression parser requires explicit multiplication. Writing 2*n makes the operation unambiguous and helps the calculator validate the expression safely.

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