Difference between 0+0+0+0...infinite times and 0 multiplied by infinity.

Question

The measure/size of set I came across that the measure of a set that is formed by intersection of countably infinite disjoint sets of measure 0 each, will be the sum of the measures of all the sets = 0+0+0+0+...(infinite terms). This can be calculated by taking the limit of the sequence of partial sums. The sequence of partial sums will be 0,0,0,... . So, from the limit of the sequence of partial sums, 0+0+0+0+... = 0. My doubt is that shouldn't 0+0+0+... = 0.$\infty$ and therefore indeterminate?

Edit 1: I'm not concerned with the limit in this case, but the actual value of 0+0+0+... and how is it different from 0.$\infty$.

Edit 2: How appropriate is it to use the limit of a function at a point instead of the function value at that point? The function need not be smooth or continuous.

Answer 1

You basically answered your question. There are no ambiguous terms in math like adding something infinite amount of times. You have to define what you mean by infinity. In this case, the infinite sum is just a limit of partial sums, by definition. So it has to be $0$.

Answer 2

Neither make sense as arithmetic expressions (no infinite number of terms, $\infty$ is not a number). In that sense they are "the same".

In analysis, an "infinite sum" is understood as a limit: $$ \sum_{k \ge 1} a_k = \lim_{N \to \infty} \sum_{1 \le k \le N} a_k $$ so your first sum would be $0$. But again, $\infty$ is not a number, so $0 \cdot \infty$ makes no sense. Closest would be some limit: $$ \lim_{x \to \infty} f(x) \cdot g(x) $$ where $\lim_{x \to \infty} f(x) = 0$ and $\lim_{x \to \infty} g(x) = \infty$. But in that case the limit could be anything at all, er even don't exist.

Answer 3

When you say the sum $0+0 + . . . + 0$, I assume you're talking about $\sum_{n=1}^\infty 0$. Since infinite summation is defined in terms of limits, the indeterminate form $0 \cdot \infty$ doesn't come up.

In other words, the reason that $0 \cdot \infty$ is indeterminate is that $\lim_{h\to\infty}h\cdot0\neq\lim_{h\to0}\infty\cdot h$. If we defined $0\cdot\infty$ as anything but indeterminate, it would imply that $\pm\infty=0$, since the first limit is zero and the second is (negative) infinity.

Since the infinite sum is equal to the first limit, it unambiguously comes out to $0$.

Answer 4

$\boldsymbol{0\cdot\infty}$

There is no real number $\infty$, so the product $0\cdot\infty$ has no meaning as a real number. When we talk about $0\cdot\infty$ being indeterminate, we are talking about $$ \lim_{n\to\infty}a_n\cdot b_n\tag1 $$ where $\lim\limits_{n\to\infty}a_n=0$ and $\lim\limits_{n\to\infty}b_n=\infty$.

As mentioned earlier, there is no real number $\infty$. What we mean by $\lim\limits_{n\to\infty}b_n=\infty$ is that for any $M\gt0$, there is an $N$ so that for all $n\ge N$, we have $b_n\ge M$; that is, $b_n$ can be made as large as one wants by choosing $n$ large enough.

To see why this is called indeterminate, for any real $r$, consider the sequences $$ a_n=\frac rn\quad\text{and}\quad b_n=n\tag2 $$ In each of these cases, we have $$ \lim_{n\to\infty}a_n\cdot b_n=\lim_{n\to\infty}r=r\tag3 $$ That is, knowing $\lim\limits_{n\to\infty}a_n=0$ and $\lim\limits_{n\to\infty}b_n=\infty$ does not tell us what $\lim\limits_{n\to\infty}a_n\cdot b_n$ is.

In conclusion, $0\cdot\infty$ is shorthand for a class of limit problems.

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$\boldsymbol{0+0+0+0+\ldots}$

This can be written as $$ \sum_{k=1}^\infty0=\lim_{n\to\infty}\sum_{k=1}^n0=\lim_{n\to\infty}0=0\tag4 $$ This is one particular series limit.