(call this poor taste, but whoever composed it was prettttty good!)
Once upon a time, there was a pretty polynomial. One day,she was strolling across a field of vectors when she came to the boundary of a singularly large matrix. Polly was convergent, and her mother made it an absolute condition that she never enter such an array without her brackets on. Polly, however,had changed her variable that morning and was feeling particularly linearly behaved, so she ignored this condition on the basis that it was insufficient and made her way amongst the complex elements. Rows and columns closed in on her from all sides. Tangents approached her apace. She becme tensor and tensor. Quite suddenly two branches of a hyperbola touched her at a single point. She oscillated violently, lost her sense of directrix, and went completely divergent. As she reached a turning point, she tripped over a square root that was protruding from the ground, and plunged headlong down a steep gradient. 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 co-ordinates, a singular expression crossed his face. He wondered, was she still convergent? He decided to intgrate improperly at once. Hearing a common fraction behind her, Polly rotated 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.
"Arcsinh!"she gasped.
"Ho, ho," he said, "what a symmetic little asymptote you have. I can see thet your angles have a lot of secs."
"Oh! Sir," she protested,"keep away from me. I haven't got any brackets on."
"Calm yourself, my dear," said our suave operator.Your fears are purely imaginary."
"I, i?" she thought, "perhaps he's not but homologous"
"What order are you?" the brute demanded.
"Seventeen"
Curly leered. "I suppose you've never been operated on?"
"Of course not," Polly replied quite properly.
"I'm absolutely convergent."
"Come, come," said Curly."Lets go off to a decimal place I know and I'll take you to the limit!"
"Never!" gasped Polly.
"Abscissa!" he swore, using the vilest oath he knew. His patience was gone.
Coshing her over the co-efficient 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, poor Polly! The algorithmic method was now her only hope. She felt his hand tending to her asymptotic limit. Her convergence would soon be gone forever. There was no mercy, for Curly was a heavy-side operator. Curly's radius squared itself,Polly's loci quivered. He integrated by parts. He integrated by patial fractions. After he co-factored, he performed Runge-Kutta on her. The complex beast even went all the way around and did a contour integration. What an indignity - to be multiply connected on her very first integration! Curly went on operating until he satisfied he hypothesis; then he exponentiated and became completely orthogonal.
When Polly got home that night her mother noticed that she was no longer piece-wise 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 over to L'Hospital 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 expressions convergent, then never allow them a single degree of freedom.