Integral Test Calculator | Improper Integral Convergence

📖 Tutorial | Integral Test

1. Theorem

If f(x) is positive, continuous and decreasing on [a,∞) and a_n=f(n), then Σ a_n and ∫_a^∞ f(x)dx share the same convergence behavior: the integral converges iff the series converges.

2. Symbols

f(x)	function matching the term
a	start of integral/sum
b	large upper bound
∫_a^∞ f	improper integral

3. How it works

Require f positive, continuous, decreasing;
Numerically integrate ∫_a^b f(x)dx by Simpson;
Also compute the tail ∫_b^{4b} f(x)dx;
A tiny tail means the integral converges, so does the series.

4. Steps

Enter f(x), e.g. 1/(x^2);
Set a=1, b=1000;
Click Calculate;
Increase b to confirm the verdict is stable.

5. Example

Example: Σ 1/n².
Solution: f(x)=1/x², ∫_1^1000≈0.999, tail ∫_1000^4000≈0.00075 is tiny; the integral converges, so the series converges.

6. Pitfalls

f must be positive, continuous and decreasing, else the theorem does not apply;
Convergence of the integral does not give the series sum;
b must be large enough for a reliable tail check.

❓ FAQ | Integral Test

What are the conditions?

f positive, continuous and decreasing on [a,∞), with a_n=f(n).

Does the integral equal the series sum?

No, it only shares convergence; the integral gives an order of magnitude.

When does a p-series converge?

Σ1/n^p converges for p>1 and diverges for p≤1.

Why compute the tail?

The upper limit is infinite; a tiny tail means the remaining integral is negligible.

What if f is not decreasing?

The test does not apply directly; check eventual monotonicity or use another test.