Can anyone show me how we can multiply two absolutely converging series without actually doing the multiplication term by term ? I read it in the book and couldn't understand much . The author says the theory is from advanced calculus and doesn't really explain a lot . Any help will do.
Well, you can multiply the results of what they converge to together rather than the series themselves. But if you want to work with expressions in series form I don't see how you can get around multiplying things out term by term.
I'm not sure how advanced calculus can help you here. I'm willing to bet though that the theorem just says you're allowed to multiply them out term by term - that doing so doesn't change the answer. This wouldn't actually help you do the multiplication though.
If you want to see proof of any of these, let me know. Some are simple, others a bit more complex.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."