## Naive Infinite Series

1. We had introduced finite series using sigma notation. Using it, a polynomial of degree $n$ may be written as
(1)$\displaystyle p(x)=\sum^{n}_{k=0}c_{k}x^{k}$
where $c_{k}$ are the coefficients. The polynomial looks like $p(x)=c_{0}+c_{1}x+\dots+c_{n}x^{n}$.

The question: what happens to a polynomial with “degree $\infty$”?

Terminology. We refer to expressions of the form $\displaystyle\sum^{\infty}_{k=0}c_{k}x^{k}$ as a “Power Series”.

Problem: given a generic power series, will it give us some closed-form expression?

2. Consider the geometric series
(2)$\displaystyle f(x)=\sum^{\infty}_{k=0}x^{k}$
which satisfies
(3)$\displaystyle xf(x)=\sum^{\infty}_{k=0}x^{k+1}$
and
(4)$\displaystyle 1+xf(x)=1+\sum^{\infty}_{k=0}x^{k+1}=f(x)$.
Thus rearranging terms we find
(5)$\displaystyle 1=(1-x)f(x)$.
Divide both sides by $(1-x)$ yields
(6)$\displaystyle \frac{1}{(1-x)}=f(x)$.
The entire “proof” has been symbolic manipulation. Do we believe it?

Lets consider $f(1/2)$, we see that the first $n$ terms in the series is
(7)$\displaystyle 1+\frac{1}{2}+\frac{1}{2^{2}}+\dots+\frac{1}{2^{n}}=\frac{2^{n+1}-1}{2^{n}}=2-2^{-n}$.
We see as $n\to\infty$, we get the series converges to
(8)$\displaystyle f(1/2)=2$.
That’s precisely $1/(1-[1/2])=1/(1/2)=2$.

What about $f(2)$? We see that it converges to $-1$. Huh?! Should we believe
(9)$\displaystyle 1+2+2^{2}+2^{3}+\dots = -1$
really is true? On the left hand side, we have purely positive numbers. How can it sum to a negative number?

Obviously we have some problems, and it appears treating power series “as if” they were polynomials is not kosher. Power series have a different, more restrictive, set of rules compared to polynomials.

3. Treating series “as if” they were polynomials can be done, but we call such a series a formal power series. The “formal” is jargon for “pretend it behaves like a polynomial”.

Remark. Historically, Euler among others treated infinite series formally, reasoning with them as if they were polynomials. Euler is a kindly saint among mathematicians, and I do not wish to belittle him: Euler’s fearlessness embodies the Biblical passage “the godly are as bold as lions” (Proverbs 28:1). Such fearless handling of infinite series is historically called the “Generality of Algebra“.

This led to such absurd results as $1 + 2 + 2^{2} + 2^{3} + \dots = -1$, as noted. But it also led to the correct solution for convergent series. We will revisit these techniques later, but first we will cover the modern rigorous treatment of series.

4. How do we tell if a series really converges? Well, one approach is to consider partial sums
(10)$\displaystyle s_{n}=\sum^{n}_{k=0}c_{k}$
as a sequence. If the sequence converges, then we just say the series is
(11)$\displaystyle \sum^{\infty}_{k=0}c_{k}=\lim_{n\to\infty}s_{n}$.
Now our problem becomes: how do we study the convergence or divergence of sequences?