Thread: Summation

Somewhere I encountered a summation in which the upper and lower bounds were infinity and negative infinity, respectively. I've looked but can't find anything about that online... I asked my Calc II teacher about it today and he that kind of thing would probably be covered in an "advanced calculus" course. Would any of you knowledgeable mathematical people care to inform me a little about this type of summation? And so you know, I'm familiar with infinite series, so I have a little background in the idea of summations...

This sort of summation isn't much different than infinite series, although there are important questions about convergence. One might called this a two-tailed series, while normal series would be one-tailed. Anyway... With a normal infinite series, it's natural to define the value of the series to be the limit of the partial sums, i.e. the value is the limit of the sums:

s_n = a₁ + ... + a_n

With a two-tailed series, you need to be more careful. There are a couple of ways of handling this that I can think of off the top of my head:

1. Absolute convergence: This isn't so much a way of dealing with series as it is a property which guarantees convergence. This basically means the same thing as absolute convergence for regular series--it doesn't matter which order you sum things up, just so long as you pick an order and the sum of the absolute values of the term converges, you're good.

2. One two-tailed series = two one-tailed series: Break your series in half--say, if the terms of a_n, where n ranges over all integers, considering the one-tailed series over a_n for n ≥ 0 and then the series over b_-n = a_n for n < 0. If each of these converges, then your two-tailed series is the sum of these two series.

3. Two-sided partial sums: For integers m < n, let s_m,n be the sum a_m + ... + a_n. Then define the value of the series to be the limit as m goes to negative infinity and n goes to positive infinity of the s_m,n. Note: this isn't assuming that m and n go to their respective infinities at the same rate. Rather, you have to consider all possible ways that m and n approach these limits. (This method may actually be equivalent to the previous method, but I haven't thought too hard about the details.)

Aside from these issues, two-tailed series really work the same as normal one-tailed series.

What good are two-tailed series? A lot of famous functions are defined by two-tailed series--for example, the theta function used to prove the functional equation of the Riemann zeta function:

∑exp(πin²z)

where the sum is on n from negative infinity to infinity.

Originally Posted by Chemboy

Somewhere I encountered a summation in which the upper and lower bounds were infinity and negative infinity, respectively. I've looked but can't find anything about that online... I asked my Calc II teacher about it today and he that kind of thing would probably be covered in an "advanced calculus" course. Would any of you knowledgeable mathematical people care to inform me a little about this type of summation? And so you know, I'm familiar with infinite series, so I have a little background in the idea of summations...

There is infinity however I only see it based on a positive infinity. In other words you could owe an infinite amount of money. But the amount, the money, is a positive thing. You owing it is considered negative, pun intended.

As far as an x,y chart, you set an imaginary origin that may or may not correspond to a real point. Only the one quadrant the upper right is positive all other quadrants are negative. This is a sort of location kind of thing. And can go out to negative or positive infinity.

Infinity usually means that a man cannot fathom it by actual useful, meaningful use of the word.



Sincerely,


William McCormick