Once upon a time pretty little 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. Poll however, who had changed her variables that morning and was feeling particularly badly behaved, ignored these conditions on the ground that they were unnecessary, and made her way amongst the complex elements.
Rows and columns enveloped her on both 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. 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 alone, apparently in a non-Euclidian space.
She was being watched however. That smooth operator, Curly Pi, was lurking inner product. As his eyes devoured her curvilinear co-ordinates, a singular expression crossed his face. Was she still convergent, he wondered. He decided to integrate at once.
Hearing a vulgar fraction behind her, Polly turned round and saw Curly Pi approaching with his power series extrapolated. She could see at once by his degenerate conic and his dissipative 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 absolutely 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 said.
"Of course no," Polly exclaimed indignantly. "I'm absolutely convergent".
"Come, come," said Curly, "lets 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 started at her significant places and began smoothing her points of inflection. Poor Polly, all was up. She felt his digit tending to her asymptotic limit. Her convergence was gone for ever.
There was no mercy, for Curly was a Heavyside operator. He integrated by partial fractions. The complex beast even went all the way round and did a contour integration. What an indignity. To be multiply connected at 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 was 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 the story is this: If you want to keep your expressions convergent, never allow them a single degree of freedom.