Little Polly Nomial

Once upon a time (1/t) a pretty Polly Nomial was strolling across a
field of vectors when she came to the edge 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 was
feeling particularly badly behaved, ignored this condition on the
grounds that it was unsufficient and made her way in amoungst the
complex elements.

Rows and columns enveloped her on all sides. Tangents approached her
surface. She became tensor and tensor.  Quite suddenly, three branches
of a hyperbola touched her at a single point. She oscillated
violently, lost all sense of directrix, and went completely
divergent. As she reached a turning point she tripped over a square
root which was protruding from the erf and plunged headlong down a
steep gradient. When she was differentiated once more she found
herself apperently 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. Was she still
convergent, he wondered. He decided to integrate improperly at once.

Hearing a vulgar fraction behind her Polly turned around and saw Curly
Pi approaching with his power series extrapolated. She could see at
once by his degenerate conis and his dissapative terms that he was
bent on no good.

"Eureka!" she gasped. 

"Ho ho!" he said. "What a symmetric little polynomial you are. I can
see you're bubbling over with 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 homogeneous then?"

"What order are you?" the brute demanded. 

"Seventeen," replied Polly. 

Curly leered, "I suppose that you've never been operated on yet?" he said. 

"Of course not," Polly cried indignantly. "I'm absolutely convergent." 

"Come, come," said Curly. "Let's go off to a decimal place I know and
I'll take you to the limit."

"Never!" gasped Polly. 

"EXCHLF!" 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 stated at her
significant places and began smoothing her points of inflection. Poor
Polly. All was up. She felt his hand tending to her asymptotic
limit. Her convergence would soon be gone forever.

There was no mercy, for Curly Pi was a Heavy side operator. He
integrated by parts. He integrated by partial fractions. The complex
beast even went all the way and did a contour integration. What an
indignanty! To be multiply connected on her first integration. Curly
went on operating until he was absolutely, and completely orthogonal.

When Polly got home that evening her mother noticed that she had been
truncated in several places. But it was too late to differentiate
now. As the months went by Polly increased monotonically. Finally she
generated a small but pathological function which left surds all over
the place until she was driven to distraction. The moral of our sad
story is this: if you want to keep your expressions convergent, never
allow them a single degree of freedom.