Once upon a time (1/t) a pretty little Polynomial 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 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 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 differential once more, she found herself, apparently, 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 coordinate a circular expression crossed his face. "Was she convergent?", he wondered. He decided to integrate improperly at once. Hearing a vulgar fraction behind her, Polly turned round and saw Curly Pi approaching her with his power series extrapolated. She could see at once by his degenerate conic and his dissipate 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."
"O Sir", she protested, "keep away from me. I haven't got my brackets on."
"Calm yourself, my dear", said our smooth operator.
"i, i" she thought, "perhaps he's homogeneous then?"
"What order are you?", the brute demanded.
"17", replied Polly.
Curly leered, "I suppose you've never been operated on before?" he said.
"Of course not", Polly replied indignantly, "I'm absolutely convergent".
"Come, come", said Curly, "let's take off to a decimal place I know, and I'll take you to the limit."
"Never" gasped Polly.
"P1000", he swore, using the vilest oath he knew. His patience was gone.
Coshing her over the coefficient with a lot until she was powerless, Curly removed her discontinuities. He stared at her significant places and began smoothing her points of inflection. Poor Polly was all up. She felt her hand tending to her asymptotic limit. Her convergence would soon be gone for ever.
There was no mercy, for Curly was a Heavy side operator. He integrated by partial fractions. The complex beast even went all the way round and did a contour integration; Curly went on until he was absolutely 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: never, if you want to keep your expressions convergent, allow them a single degree of freedom.
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