A Series might be a number or a sequence – Is there a better notation?!

Question

Take the expression $\sum_{k=1}^\infty a_k$. Sometimes this expressions refers to the sequence of partial sums $\left(\sum_{k=1}^n a_k\right)_{n\in\mathbb N}$ and sometimes to the limit of this sequence $\lim_{n\to\infty} \sum_{k=1}^n a_k$ (when this limit exists). For example in the expression

The series $\sum_{k=1}^\infty \left(\frac 12\right)^n$ converges.

the term $\sum_{k=1}^\infty \left(\frac 12\right)^n$ is a sequence. In the expression

It is $\sum_{k=1}^\infty \left(\frac 12\right)^n = 1$.

the same expression $\sum_{k=1}^\infty \left(\frac 12\right)^n$ is a real number.

For me this situation is unsatisfactory. Any expression in mathematics shall have an unique interpretation and shall not be ambiguous in its meaning. For example we explain the necessary of the expression $\sum_{k=1}^n a_k$ because it might not be clear how to interpret an expression like $a_1 + a_2 + a_3 + \ldots + a_n$ (i.e. how to fill the dots). So there also shall not be an ambiguity in $\sum_{k=1}^\infty a_k$.

My question: Is there a textbook or script which differs the two interpretations of $\sum_{k=1}^\infty \left(\frac 12\right)^n$, i.e the sequence of partial sums $\left(\sum_{k=1}^n a_k\right)_{n\in\mathbb N}$ and $\lim_{n\to\infty} \sum_{k=1}^n a_k$? I am looking for a textbook, where the author omit the expression $\sum_{k=1}^\infty \left(\frac 12\right)^n$ or where he uses this expression only for the partial sum sequence or only for the limit.

Reason for my question: I am looking for a clear notation I can use for my textbook. So also suggestions for a better notation are welcome ;-)

Answer 1

The usual notation for a sequence is $\langle a_n \rangle$, or whatever shape suits your fancy for the enclosing brackets, and if you want you can have state the index like $\langle a_n \rangle_{n=0}^\infty$, or similar. Then, we use $a_n$ as the $n$th term of the sequence, and we can speak of $\displaystyle \lim_{n \to \infty} a_n$ as a number, if $\langle a_n \rangle$ converges.

As series are sequences, you can state in your textbook that $\sum a_n$ is always a number, (or a power series, etc,) if it exists, and if you wish to speak of a sequence whose $n$th term is $\sum_{i=0}^n a_i$, the notation would be $\displaystyle \left \langle {\sum_{i=0}^n a_i }\right \rangle_{n\ge 0}$. Though this at first glance seems unconventional, the meaning should be immediately obvious as long as you remember that series are sequences of partial sums.

If this makes you feel uncomfortable and you wish it to look more "standard", you can write:

$$\displaystyle \left \langle {S_n}\right \rangle_{n\ge 0}; \quad S_n = \sum_{i=0}^n a_i$$

Answer 2

First I would like to point out that the expression $\sum_{k=1}^\infty (\frac{1}{2})^n$ does not make sense to me at all. Anyway, it seems like you are writing a textbook or lecture notes. As a student who struggled during my first real analysis course, I would suggest the following. First we should clearly define what $\sum_{k=1}^{\infty}a^k$ means. Notice that if we replace the $\infty$ with some real number (say $n$) then there should be no ambiguity. Given that it is infinity, some students might get confused and believe that this expression is always a real number, which is not the case. So it would be helpful if you warn them that even though mathematicians write this all the time, it might not be meaningful at all if it doesn't converge.

In terms of the two different interpretations, here is how I would approach it. Usually when we talk about partial sum, we are talking about the expression $\sum_{k=1}^{n} a_k$. Notice that here $n$ has to be a natural number. If we accept that as part of our definition, then the expression $\sum_{k=1}^\infty a_k$ is not a partial sum. Then there should be no ambiguity here. In other word, I don't think using the symbol $\sum$ has any problem. Instead the tricky part is the upper limit of the $\sum$. Is it a natural number? Or is it infinity? As long as we can explain the vital difference here, we should be good.