In honor of Robert Payne, who (I have just noticed) is our second professional mathematician, I offer the following fable, its wit justifying its length:
Pretty Little Poly Nomial and Curly Pi
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, ever 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, and she became tenser and tenser. Quite suddenly, two branches of a hyperbola touched her at a single point. She oscillated violently, became unstable, lost all sense of directrix, tripped over a square root that was protruding from the erf, and plunged headlong down a steep gradient. She was completely divergent by the time she reached the turning point. 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 convergent? He decided to integrate improperly at once.
Hearing a common fraction behind her, Polly rotated and saw Curly Pi approaching with his lower series extended. She could see at once his degenerate conic and his dissipative terms, and knew he was irrational. "Arcsinh!" she gasped.
"Hey, what's your sine?" he asked. "What a symmetric set of asymptotes you have!"
"Stay away from me!" she protested. "I haven't got any brackets on!"
"Calm yourself, my dear!" said the smooth operator.. "Your fears are purely imaginary."
"i, i, ..." she thought, "Perhaps he's not normal, but homologous."
"What order are you?" the brute suddenly demanded.
"Seventeen," replied Polly.
Curly leered, "I suppose you've never been operated upon?"
"Of course not. I'm absolutely convergent!" Polly replied quite properly.
"Come on," said Curly: "Let's go to decimal place I know of, and I'll take you to the limit."
"Never!" gasped Polly..
"Abscissa!" he swore a violent oath. 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 her points of inflection. Poor Polly Nomial! The algorithm method was now her only hope. She felt him approaching her asymptotic limit. Her convergence would soon be gone forever. There was no mercy; Curly was a heavy side operator. His radius squared itself and Polly's loci quivered. He integrated by parts. He integrated by partial fractions. After he cofactored, he performed Runge-Kutta on her. He even went all the way around and did a contour integration. Curly went on operating until he satisfied her hypotheses, 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 too late to differentiate now. As the months went by, Polly's denominator increased monotonically. Finally, they took her 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 this tale is:
"If you want to keep your expressions convergent, never allow them a single degree of freedom."