Originally Posted by Got in an Email

Impure Mathematics
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To prove once and for all that math can be fun, we present:

Wherein it is related how that paragon of womanly virtue, young Polly
Nomial (our heroine) is accosted by that notorious villain Curly Pi,
and factored (oh horror!!!)

Once upon a time (1/t) pretty little Polly Nomial was strolling across
a field of vectors when she came to the boundary of a singularly large
matrix. Now Polly was convergent, and her mother had made it an
absolute condition that she must never enter such an array without her
brackets on. Polly, however, who had changed her variables that
morning and was feeling particularly badly behaved, ignored this
condition on the basis that it was insufficient and made her way in
amongst the complex elements. Rows and columns closed in on her from
all sides. Tangents approached her surface. She became tensor and
tensor. Quite suddenly two branches of a hyperbola touched her at a
single point. She oscillated violently, lost all sense of directrix,
and went completely divergent. She tripped over a square root that was
protruding from the erf and plunged headlong down a steep gradient.
When she rounded off once more, she found herself inverted, apparently
alone, in a non-Euclidean space.

She was being watched, however. That smooth operator, Curly Pi, was
lurking inner product. As his eyes devoured her curvilinear
coordinates, a singular expression crossed his face. He wondered, "Was
she still convergent?" He decided to integrate properly at once.

Hearing a common fraction behind her, Polly rotated and saw Curly Pi
approaching with his power series extrapolated. She could see at once
by his degenerate conic and dissipative that he was bent on no good.
"Arcsinh," she gasped.
"Ho, ho," he said, "What a symmetric little asymptote you have I can
see your angles have lots of secs."
"Oh sir," she protested, "keep away from me I haven't got my brackets on."
"Calm yourself, my dear," said our suave operator, "your fears are
purely imaginary."
"I, I," she thought, "perhaps he's not normal but homologous."
"What order are you?" the brute demanded. "Seventeen," replied Polly.
Curly leered "I suppose you've never been operated on."
"Of course not," Polly replied quite properly, "I'm absolutely convergent."
"Come, come," said Curly, "let's off to a decimal place I know and
I'll take you to the limit."
"Never," gasped Polly.
"Abscissa," he swore, using the vilest oath he knew.
His patience was gone. Coshing her over the coefficient with a log
until she was powerless, Curly removed her discontinuities. He stared
at her significant places, and began smoothing out her points of
inflection. Poor Polly. The algorithmic method was now her only hope.
She felt his digits tending to her asymptotic limit. Her convergence
would soon be gone forever.

There was no mercy, for Curly was a heavyside operator. Curly's radius
squared itself; Polly's loci quivered. He integrated by parts. He
integrated by partial fractions. After he cofactored, he performed
runge-kutta on her. The complex beast even went all the way around and
did a contour integration. What an indignity - to be multiply
connected on her first integration. Curly went on operating until he
completely satisfied her hypothesis, then he exponentiated and became
completely orthogonal.

When Polly got home that night, her mother noticed that she was no
longer piecewise continuous, but had been truncated in several places.
But it was to late to differentiate now. As the months went by,
Polly's denominator increased monotonically. Finally she went to
L'Hopital and generated a small but pathological function which left
surds all over the place and drove Polly to deviation.

The moral of our sad story is this: "If you want to keep your
expressions convergent, never allow them a single degree of freedom."