# Alephs of Sums@infinity

1. Oct 1, 2004

### bomba923

How do I represent the quantity of terms for a summation with an upper limit approaching infinity? In the attached image, the limit is obviously infinity, but the quantity of terms following the summation is countable! (as the upper limit approaches infinity, the quanitity of terms approaches a countable infinity (natural numbers)). If x can be reals, then the expression would not work, as reals would cause an uncountable infinity. However, x can be either rational or irrational, but i have a problem when i set the domain to BOTH rational and irrational at the same time in the same domain. It seems that x can be either, but not both at the same time.(see the uploading image)

Well, i'm just wondering, but what is really the domain of the expression---what set of numbers does x belong to?? In addition, how could i represent the quantity of terms following the sigma sum; what aleph would i use? In addition, how would i write the upper limit as a countable infinity? (not with a sideway eight, cuz i want to show that it approaches a countable infinity. Maybe the aleph-upsilon, or just the upsilon symbol would do?)

*this seems a simple problem for which i couldn't solve for some reason//maybe lack of knowledge of something....worse.. However, i seemed to have chosen this problem to present to my class (i'm a high-school junior, but this was just a small extra-credit problem), so i researched some material over the internet. unfortunately, it wasn't enough and the info seemed too vague)

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2. Oct 1, 2004

### matt grime

Your zip file doesn't contain anything i can look at, try uploading as text or something similar.

hte way to write sums with a (countable) number of terms is

$$\sum_{n=1}^{\infty}a_n$$

the rest of your post doesn't make sense to me, especially since i can't see your file.
You do not use alephs like this, that is not what they are there for in analysis. you cannot have a sum of uncountably many non-zero real numbers anyway, and summation is an ordered process, it's important to remember that, aleph's are cardinals, not ordinals.

sums *are* written with the sideways 8 if you are summing a series.

your use of quantity is vague and i cannot decide what you are referring to. quantity is a bad word.

do you mean the cardinality of the set of terms in the series?

3. Oct 1, 2004

### Hurkyl

Staff Emeritus
First off, the alephs have nothing to do with this problem. Alephs are only relevant when you're interested in the "number" of elements in a set.

The second thing to note is that the number of the terms in the summation is finite! This is not an infinite sum: the upper and lower bounds are always (finite!) real numbers. What we actually have here is a subtle concept that could be named "unbounded but finite"; there is no (finite) bound on how many terms are in the sum, but the number of terms is always finite.

Because the limit is as x approaches 0 from the right, the domain for x may be taken to be your favorite interval of the form (0, a) where a is a positive real number, but less than c (because the problem requires x < c).

Finally, you're making the problem harder than it looks. What would you do if you weren't taking a limit? Do that, then worry about the limit.

Last edited: Oct 1, 2004
4. Oct 1, 2004

### Hurkyl

Staff Emeritus
Oh, matt, the problem is:

$$\lim_{x \rightarrow 0^+} x \sum_{n=1}^{\frac{c}{x}} n$$

Where c is positive.

5. Oct 2, 2004

### bomba923

Can i edit posts that i made some time ago?? Like i can't edit my post about the infinite sum thing...how can i edit posts that i made some time ago?? I can edit posts that i make right now, but how come i can't edit my earlier posts?