Foundations — Term-by-term differentiation and integration of power series

This page assumes nothing. Every letter, symbol and picture the parent note parent topic leans on is built here from the ground up, in an order where each block rests on the one before it.

0. The alphabet we are about to use

Before any formula, here is the plain-words meaning of each symbol. We earn each one properly in its own section below.

1. $\sum$, the counter $n$, and $\infty$ — "add up a list"

Read it out loud as: "for $n$ from $0$ to $3$, add up each $t_n$." Each $t_n$ is one term — one item on the shopping list.

Look at Figure s01. The cyan dots are the running total after $1,2,3,\dots$ terms of $\frac{1}{2}+\frac{1}{4}+\cdots$. The white arrows show each dot creeping toward the amber dashed line at height $1$ — a picture of "homing in" (convergence).

2. The weights $a_n$, the input $x$, and the function $f$

Now we make each term depend on a number $x$ we get to choose.

A simple example: if every $a_n=1$, the terms are $1,x,x^2,x^3,\dots$ and $f(x)={\sum }_{n=0}^{\infty }{x}^n=1+x+x^2+x^3+\cdots$ This is the Geometric series — the friendliest power series there is, and the parent note builds everything from it.

3. The building block $(x-c{)}^n$ and the centre $c$

So the general power series in the parent note is $f(x)={\sum }_{n=0}^{\infty }{a}_n(x-c{)}^n=a_0+a_1(x-c)+a_2(x-c{)}^2+\cdots$

where $f(x)$ (from §2) is the number the whole sum lands on when you feed in $x$.

Look at Figure s02. The amber curve is the true total $\frac{1}{1-x}$; the cyan/white curves are what you get by stopping after a few terms of $1+x+x^2+\cdots$. Each extra term hugs the amber curve over a wider stretch — an infinite polynomial is the smooth function, in the limit.

4. Absolute value $|\cdot |$ and the radius $R$

Plug in a big $x$ and the powers $(x-c{)}^n$ explode; the running totals run off to infinity (diverge). Plug in a small $x$ and the terms shrink; the running totals settle (converge). There is a cutoff distance — and distance ignores direction, which is what absolute value is for.

Look at Figure s03. The cyan band is the safe zone of half-width $R$ around the amber centre dot $c$; the white squares mark the two edges $c\pm R$, flagged "test by hand."

5. $\mathrm{limsup}$ — reading the radius off the coefficients

How do we find $R$? The Cauchy–Hadamard theorem does it from the weights alone: $\frac{1}{R}={\mathrm{limsup}}_{n\to \infty }|a_n{|}^{1/n}.$

Two new pieces here.

Look at Figure s04. The cyan dots are a wobbly list of $|a_n{|}^{1/n}$ values; the amber dashed line is the $\mathrm{limsup}$ — the ceiling the peaks keep returning to even though individual dots dip below it.

The parent's whole radius proof rests on one fact: multiplying weights by the counter $n$ barely nudges this $n$-th-root scale, because $n^{1/n}\to 1$. We can sanity-check that number: even at $n=100$, $100^{1/100}\approx 1.047$ — already almost $1$.

6. $\frac{d}{dx}$ and $f^{\prime }$ — the slope machine

Look at what that rule does to a term: it multiplies the weight by $n$ and knocks the power down by one. That is precisely why the parent's differentiated series reads $\sum n{a}_n(x-c{)}^{n-1}$ — every term got hit by the same rule. The $n=0$ term (a constant) has slope zero and vanishes, which is why the sum restarts at $n=1$.

7. $\int$ — the area machine

The $+C$ (the constant of integration) is an unknown constant: many functions share the same slope-pattern, differing only by a fixed vertical shift, so integration leaves that shift undetermined until you pin it with one known value (like $f(0)$). This is exactly why the parent fixes $C$ using $\arctan 0=0$ and $\ln (1+0)=0$.

Applied to every term, this gives the parent's integrated series $C+\sum \frac{a_n}{n+1}(x-c{)}^{n+1}$: each weight divided by $n+1$, each power bumped up one.

Prerequisite map

Equipment checklist

Test yourself — say the answer before revealing.