I am considering two sets of symbols. Are these two sets different or the same? Why?
\begin{align}(1)\qquad&\lim_{n\to\infty}\sum_{j=1}^n\\ (2)\qquad&\sum_{j=1}^\infty\end{align}
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1They are the same. – 76david76 Apr 19 '17 at 05:28
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- is the definition of 2.
– Masacroso Apr 19 '17 at 05:29 -
I am seeing a subtle difference between the two, but wondering if it is noteworthy. 1) involves n approaching infinity but never actually reaching infinity. 2) involves n actually being equal to infinity. So is approaching infinity the same as being infinite? Do speeds at which infinity is approached matter at all in the context of summation? – Math12345 Apr 19 '17 at 05:37
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2You are asking the right questions. In this case it's the answer that is hinky. Actually taking and adding an infinite sum is not only impossible but meaningless. We can never do it. So what we would think $\sum_{i=1}^{\infty} $ should mean... it doesn't. Instead we define that whenever we write these symbols on ink on paper "$\sum_{i=1}^{\infty}$" we define it to mean $\lim_{n\rightarrow\infty}\sum_{i=1}^{n} $. That is just what it means. We can't argue it is wrong, because that is how it is defined. – fleablood Apr 19 '17 at 06:41
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Does the second interpretation listed below (i.e. limit of the sequence of partial sums) work when the upper limit of the sum is a finite value such as 0 or 1? This case would involve considering the limit of $n$ as $n$ approaches a finite value. The limit evaluates to the finite value. However the subtlety mentioned above becomes clearer in this finite case since the upper limit of the sum can be a finite value, but it can also be the value of the limit. So my question becomes, does the value of a limit have the same meaning as the same value that is not the result of taking a limit? – Math12345 Apr 20 '17 at 04:11
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@Math12345: We can't have $n \to 3$ kind of scenario. The essential / fundamental ingredient of the limit operation is that there must be infinitely many values of the variable to consider. When $n \to 3$ and $n$ is a positive integer then we don't have infinitely many values of $n$ close to $3$ so it does not make sense. If $x \to 3$ and $x$ is real variable then it makes sense because there are infinitely many real numbers close to $3$. – Paramanand Singh Apr 20 '17 at 06:58
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1@Math12345: It would be best if you can get hold of "A course of Pure Mathematics" by G. H. Hardy (it should be available freely online if you search enough). It discusses exactly the questions you are asking in great detail in very simple language. I did not find any calculus/analysis book coming even slightly close to it which is suitable for beginners for self study – Paramanand Singh Apr 20 '17 at 07:01
3 Answers
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The series (2) in the form $\sum_{j=1}^{\infty}a_n$ admits two different interpretations.
It can be seen as sequence of partial sums \begin{align*} \sum_{j=1}^{\infty}a_j:=\left(a_1,a_1+a_2,a_1+a_2+a_3,\ldots\right)=\left(\sum_{j=1}^na_j\right)_{n\geq 1} \end{align*}
and it can be seen as limit of the sequence of partial sums \begin{align*} \sum_{j=1}^{\infty}a_j:=\lim_{n\to\infty}\sum_{j=1}^na_j \end{align*}
In the second case the symbols (1) and (2) have the same meaning whereby (2) is defined by (1).
Note: You can find this line of argumentation e.g. in K. Knopps Theory and application of infinite series.
Markus Scheuer
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Lets write these out carefully for some sequence of numbers $a_n$.
The first thing you wrote is $$\lim_{n\to\infty}\sum_{j=1}^na_n.$$ This is already pretty well defined (if you are comfortable with the definition of a limit of a sequence, since that is exactly what this is).
The second thing is $$\sum_{j=1}^\infty a_n.$$ This does not appear to be a limit (at least not explicitly), however we cannot actually keep adding infinitely many terms to each other, so there must be some limit involved. This hidden limit is precisely the first thing you wrote, and as mentioned in the comments the first expression is the definition of this second expression.
helloworld112358
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I guess in some formal setting the sum $$\sum_{j=1}^{\infty}()$$ is defined to be $$\lim_{n \to \infty} \sum_{j=1}^{n}()$$, how else can we take infinite sums or prove results? With the limit. Indeed if you "believe" in infinity or not the results can be positively guaranteed.
marshal craft
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