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Improper integrals

Integrals to infinity or across a singularity, defined as limits.

∫₁^∞ 1/x² dx is a limit of ∫₁^b as b → ∞, and it converges to 1; the same integral of 1/x diverges. The boundary between the two, p = 1, is the same boundary as for the p-series. The integral test is exactly this correspondence.

Àwọn ààtò àwọn ìṣàmúlò-ètò · integrate 1/x^2 dx from 1 to oo

Calculus · Definite-Integrals

Integrate x^(-2) from 1 to oo

\int_{1}^{\infty} \frac{1}{x^{2}}\, dx

Àwọn ìṣàmúlò-ètò

\int_{1}^{\infty} \frac{1}{x^{2}}\, dx
First find an antiderivative F, then evaluate F(b) − F(a).
\int \frac{1}{x^{2}}\, dx = - \frac{1}{x}
Power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) (n ≠ −1).
F(\infty) - F(1) = \left(0\right) - \left(-1\right)
Fundamental theorem of calculus: plug in the limits.
= 1
Simplify.

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Bii o ṣe le: Improper integrals

First find an antiderivative F, then evaluate F(b) − F(a).
Power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) (n ≠ −1).
Fundamental theorem of calculus: plug in the limits.
Simplify.

Àwọn Àtòjọ-ẹ̀yàn

What is the ε, δ definition actually saying?

That you can make the output as close to the limit as anyone demands (within ε) by keeping the input close enough (within δ). It replaces "approaches" with a challenge-and-response that can be checked.

Why does the harmonic series diverge when its terms go to zero?

Because the terms shrink too slowly: group them as 1/3 + 1/4 > 1/2, 1/5 + … + 1/8 > 1/2, and so on: infinitely many halves.

Diẹ̀ nínú Real Analysis

Sequences and their limits Convergence of series Taylor approximation and error Functions of a complex variable Continuity and differentiability, rigorously Sequences of functions and uniform convergence