Once upon a time, (1/T) pretty little Polly Nomial was strolling through 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 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 grounds that it was insufficient and made her way in amongst the complex elements.
Rows and columns enveloped her on all sides. Tangents approached her surface. She became tensor and tensor. Quite sudenly, 3 branches of a hyperbola touched het 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 protruding from the erf and plunged headlong down a steep gradient.
When she was differentiated once more, she found herself, 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. 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 conic and his dissipated terms, that he was up to no good.
“Eureka,” she gasped.
“Ho, ho,” he said.
“What a symmetric little polynomial you are. I can see you are 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 you’ve never been operated on yet?” he asked.
“Of course not!” Polly cried indignantly. “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. “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 stared 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 was a heavyside operator. He integrated by parts. He integrated by partial fractions. The complex beast even went all the way around and did a counter integration. What an indignity to be multiply connected on her first integration. Curly went on operating until he was absolutely and 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, 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 expression convergent, never allow them a single degree of freedom.