Mathematical Rules of Thumb, illustrated reader · Chapter 5

05Analysis

Know When Limits and Proofs Are Safe

4 demonstrations follow the chapter's rules. Choose a value, watch the figure and the numbers change, and check your prediction. Every choice is precomputed from the notebook calculations.

Ask the chapter skill

“Help me use Chapter 5 for my question. Choose a rule, check its assumptions, and show how the result changes when an input changes.”

Use math-thumb-analysis from the companion's skill package. The demonstrations below also work on their own.

Examples use constructed inputs or the book's own values, disclosed in each panel. A picture illustrates a rule; its assumptions set its scope.

1Demonstration 1 of 4

Turn convergence speed into a stopping certificate

Why does q=.9 need more steps than q=.5?

A contraction pulls every pair of points closer by the factor q each step. The fixed point is known here, so the exact error and the contraction bound can be compared.

T(x)=qx+(1−q),|xn−1|=qn T(x)=qx+(1-q),\quad |x_n-1|=q^n

Contraction q. 0<q<1 on the complete real line; x₀=0. The affine example is a special case.

Predict first. Why does q=.9 need more steps than q=.5?

Choose an example

Turn convergence speed into a stopping certificate. T(x)=q x+(1-q), starting at zero, has error qⁿ. At q=0.5, 7 steps suffice for .01 error. In this affine example the bound equals the exact error, so the two curves coincide. q must be strictly below one.
Contraction q: 0.5
Constructed teaching inputs; calculations executed locally. Supported menu choices are precomputed.

Calculated values

Contraction q
0.5
Error after 10 steps
0.000976562
Steps for error ≤ .01
7

T(x)=q x+(1-q), starting at zero, has error qⁿ. At q=0.5, 7 steps suffice for .01 error. In this affine example the bound equals the exact error, so the two curves coincide. q must be strictly below one.

Use the idea

Use rule 5.1.2 when its stated conditions fit. Compare the calculation with your own decision threshold; retain the relevant error or uncertainty.

Where the conclusion applies

0<q<1 on the complete real line; x₀=0. The affine example is a special case.

Check your understanding: Why does q=.9 need more steps than q=.5?
Each step retains a larger fraction of the error. The geometric certificate makes the required step count explicit.

Book source: Rule 5.1.2: Turn contraction rate into a fixed-point error bound. Demonstration C05-D01. Worked illustration.

2Demonstration 2 of 4

Watch a boundary layer defeat uniform convergence

What is the supremum error to the pointwise limit for finite n?

The curve approaches zero at fixed interior points while remaining one at the endpoint. The largest gap over all x (the supremum) stays 1 for every n, so no single n makes the whole curve close to the limit.

fn(x)=xn,f(x)=0(x<1),f(1)=1 f_n(x)=x^n,\quad f(x)=0\ (x<1),\quad f(1)=1

Power n. Domain [0,1]. A sampled plot cannot establish uniform convergence; the supremum argument supplies the conclusion.

Predict first. What is the supremum error to the pointwise limit for finite n?

Choose an example

Watch a boundary layer defeat uniform convergence. Increasing n suppresses xⁿ at every fixed x<1, but a boundary layer remains near one. The supremum difference from the discontinuous pointwise limit is 1 for every n; this is not uniform convergence.
Power n: 20
Constructed teaching inputs; calculations executed locally. Supported menu choices are precomputed.

Calculated values

Value at x=.9
0.121577
Value at x=1
1
Supremum error to pointwise limit
1

Increasing n suppresses xⁿ at every fixed x<1, but a boundary layer remains near one. The supremum difference from the discontinuous pointwise limit is 1 for every n; this is not uniform convergence.

Use the idea

Use rule 5.1.3 when its stated conditions fit. Compare the calculation with your own decision threshold; retain the relevant error or uncertainty.

Where the conclusion applies

Domain [0,1]. A sampled plot cannot establish uniform convergence; the supremum argument supplies the conclusion.

Check your understanding: What is the supremum error to the pointwise limit for finite n?
It is 1, approached as x tends to 1 from below. The maximum need not be attained for the supremum to be 1.

Book source: Rule 5.1.3: Demand uniform control before interchanging limits. Demonstration C05-D02. Worked illustration.

3Demonstration 3 of 4

Control a whole function series at once

Can the same finite bound be used with q=1?

A larger geometric series (a majorant) controls the remainder for every x in the stated interval, instead of checking points individually.

∑j=n+1∞|x|j≤qn+11−q,|x|≤q<1 \sum_{j=n+1}^{\infty}|x|^j\leq\frac{q^{n+1}}{1-q},\quad |x|\leq q<1

Uniform radius q. A fixed compact interval |x|≤q strictly inside the convergence radius.

Predict first. Can the same finite bound be used with q=1?

Choose an example

Control a whole function series at once. For |x|≤0.7<1, the series sum xⁿ is uniformly controlled by sum qⁿ. Its tail after exponent 10 is at most 0.0659109; the bound deteriorates as q approaches one.
Uniform radius q: 0.7
Constructed teaching inputs; calculations executed locally. Supported menu choices are precomputed.

Calculated values

Uniform |x| ceiling q
0.7
Remainder after exponent 10
0.0659109

For |x|≤0.7<1, the series sum xⁿ is uniformly controlled by sum qⁿ. Its tail after exponent 10 is at most 0.0659109; the bound deteriorates as q approaches one.

Use the idea

Use rule 5.2.1 when its stated conditions fit. Compare the calculation with your own decision threshold; retain the relevant error or uncertainty.

Where the conclusion applies

A fixed compact interval |x|≤q strictly inside the convergence radius.

Check your understanding: Can the same finite bound be used with q=1?
No. Its denominator vanishes and the geometric majorant fails. A different domain or argument is needed.

Book source: Rule 5.2.1: Use the M-test for uniform convergence of function series. Demonstration C05-D03. Worked illustration.

4Demonstration 4 of 4

Test absolute convergence before reordering a series

Can reordering 1−1/2+1/3−... change its sum?

Use the same terms ±1/n but choose the order: add a positive term while at or below the target, a negative one while above. The running sum lands on the target.

1−12+13−14+⋯=ln⁡2,∑1n=∞ 1-\tfrac12+\tfrac13-\tfrac14+\cdots=\ln 2,\quad \sum\tfrac1n=\infty

Target sum for a rearrangement. Conditionally convergent series only. If the sum of absolute values converges, every order gives the same sum.

Predict first. Can reordering 1−1/2+1/3−... change its sum?

Choose an example

Test absolute convergence before reordering a series. Adding positives while at or below 1.5 and negatives while above steers the running sum to 1.5. The terms are exactly those of the ln 2 series; only the order changed. Because the sum of |terms| diverges, order is part of the answer.
Target sum for a rearrangement: 1.5
Constructed teaching inputs; calculations executed locally. Supported menu choices are precomputed.

Calculated values

Target
1.5
Sum after 3000 terms
1.49977
Usual-order sum ln 2
0.693147

Adding positives while at or below 1.5 and negatives while above steers the running sum to 1.5. The terms are exactly those of the ln 2 series; only the order changed. Because the sum of |terms| diverges, order is part of the answer.

Use the idea

Use rule 5.2.3 when its stated conditions fit. Compare the calculation with your own decision threshold; retain the relevant error or uncertainty.

Where the conclusion applies

Conditionally convergent series only. If the sum of absolute values converges, every order gives the same sum.

Check your understanding: Can reordering 1−1/2+1/3−... change its sum?
Yes. The positive and negative parts each diverge, so any target can be reached. Absolute convergence is what makes order irrelevant.

Book source: Rule 5.2.3: Test absolute convergence before conditional behavior. Demonstration C05-D04. Worked illustration.

Bring the idea to a question of your own

Choose the relationship that answers your question, check its conditions, and compare the result with the accuracy or decision threshold you need.

The chapter skill can adapt these calculations to your inputs. It should name the assumptions, explain what the result supports, and say what still needs evidence. The chapter workbook adds a lab and three exercises with answers.