Foundations — Convergence tests for improper integrals — comparison

This page assumes you have seen nothing. Before you can judge whether ${\int }_1^{\infty }\frac{dx}{x^2+5}$ converges, you must own every mark on the page — the $\int$, the $\infty$, the $\le$, the word "limit". We build each from a picture, in the order they depend on each other.

1. Area under a curve — what $\int$ actually draws

The picture. Slice the region into thin vertical strips. Each strip sits at some $x$, has height $f(x)$, and a tiny width we call $dx$. Its area is height $\times$ width $=f(x)dx$. The stretched-S symbol $\int$ is an old-fashioned letter S for "Sum" — it means "add up all those thin strips." The $a$ and $b$ are the limits of integration: where the summing starts and stops.

Why the topic needs it. Every claim in comparison ("this area is finite / infinite") is a claim about $\int$. If $\int$ is not a picture of accumulated area in your head, none of the tests mean anything.

2. The symbols $f$, $x$, $g$ — functions as height-machines

The picture. Stand at position $x$ on the floor. Look straight up until you hit the curve. The height you climbed is $f(x)$. In this whole topic $f$ is the hard function whose area we cannot compute, and $g$ is the easy benchmark function we compare it to. They live on the same axes, so we can literally see which curve is higher.

Why the topic needs it. "Compare $f$ to $g$" means "at every position $x$, is the height of $f$ below or above the height of $g$?" That is a statement you must be able to see.

3. The inequality $0\le f(x)\le g(x)$ — one curve trapped under another

Why $0\le f$ too? The extra "$0\le$" says $f$ never dips below the floor — its area only ever grows as we widen the window, never shrinks. Look at the figure: the blue $f$ hugs the ground, orange $g$ floats above it. Because blue is squeezed into the strip between floor and orange, the blue area can never exceed the orange area. That single sentence is the entire engine of the Direct Comparison Test — pardon, of the parent's Direct Comparison Test.

Why the topic needs it. DCT is literally this inequality plus one logical step. Get the direction of $\le$ wrong and every conclusion flips.

4. The symbol $\infty$ and why we sneak up on it with a limit

The picture. Let $G(t)={\int }_1^{t}fdx$ be the area from $1$ out to the sliding wall at $t$. As you drag the wall right, more area accumulates, so $G(t)$ climbs. Two things can happen:

• It levels off toward a ceiling — the total area is a finite number. We say the improper integral converges.
• It keeps rising without a ceiling — the total area is infinite. It diverges.

Why the topic needs it. "Converges" and "diverges" are defined by this limit. The whole point of comparison is to decide which of the two red curves in the figure your unknown area follows — without computing the curve exactly.

5. Powers $x^{-p}$ and the benchmark idea

The picture. Compare $\frac{1}{x}$ (drops gently) with $\frac{1}{x^2}$ (drops steeply). The steeper one leaves less area in its long tail. There is a knife-edge exponent, $p=1$, dividing "tail area finite" from "tail area infinite":

${\int }_1^{\infty }\frac{dx}{x^p}\text{converges}⟺p>1.$

Why the topic needs it. Almost every benchmark $g$ is a power $x^{-p}$. The single fact "$p>1$ ⇒ finite, $p\le 1$ ⇒ infinite" is the yardstick every comparison measures against. See The p-integral and p-series for the full derivation the parent gives.

6. The ratio $L=\lim f/g$ — comparing growth rates instead of heights

The picture. Imagine $f$ and $g$ marching to the right. If their heights stay in a fixed proportion (say $f$ is always roughly $3$ times $g$), then wherever one's total area is finite, so is the other's; wherever one is infinite, so is the other. The ratio $L$ measures that stable proportion.

Why the topic needs it. When the inequality $f\le g$ is fiddly to prove by hand, the Limit Comparison Test lets you skip the fight: just check the dominant terms match. That is the tool behind Example 2 in the parent, where $\frac{1}{\sqrt{x}+x}$ behaves like $\frac{1}{x}$.

How these foundations feed the topic

Equipment checklist

Test yourself — cover the right side and answer out loud.

Connections

• Yeh note Hinglish mein padho →
• The p-integral and p-series
• Improper integrals — infinite discontinuities (type 2)
• Monotone Convergence Theorem
• Comparison test for infinite series
• Integral Test for series
• Absolute vs conditional convergence