Sigma notation for only even index iterations

[hquang001]

TL;DR Summary

How can i write sigma sum, for only even index interation ?

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[fresh_42]

Summary:: How can i write sigma sum, for only even index interation ?

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$f(2)+f(4)+f(6)+\ldots + f(2n)= \sum_{k=1}^{n} f(2k).$

 

[mfb]

$$\sum_{k=2,\,k~even}^{n} k$$ or similar if you need it for consistency with other sums.

$$\sum_{k=1}^{n} k = \sum_{k=2,\,k~even}^{n} k + \sum_{k=1,\,k~odd}^{n} k$$

This is more commonly done with criteria that can't be resolved by simple relabeling.

$$\sum_{k=2,\,k~prime}^{n} k$$

 

[fresh_42]

What an ugly solution! If you really want to restrict the index of the sum, then it should be
$$
\sum_{k=1}^n f(k)=\sum_{\stackrel{k=1}{k\equiv 0 (2)}}^n f(k) + \sum_{\stackrel{k=1}{k\equiv 1 (2)}}^n f(k)
$$
and for primes only
$$
\sum_{p\in \mathbb{P}} f(p)
$$
and for prime divisors
$$
\sum_{p|n} f(p)
$$
Additional text is in my opinion worse than dots.

 

[sysprog]

Like most people (I presume), I use Newton's dot notation $only$ in responses to things that include expressions that use it $-$ I think that I see it more often with Physics guys than with math guys $-$ otherwise for single variable derivatives it's LaGrange's $f', f''$, etc. (for me, only up to $f''''$ ('jounce/snap') so far $-$ I haven't had a reason for $f'''''$ ( 'crackle') or $f''''''$ ('pop')), and for multivariable or integrations, Leibniz' $\dfrac {dy} {dx}$.

 

[pasmith]

Some text is appropriate, provided it is typeset as text. If I don't know in advance that $n$ is even. I would prefer $\displaystyle\sum_{0 \leq k \leq n,\atop\text{k even}} f(k)$ produced with

\displaystyle\sum_{0 \leq k \leq n,\atop\text{$k$ even}} f(k)

rather than $\displaystyle \sum_{k=0}^{\lfloor n/2 \rfloor} f(2k)$ or any other notation which means "$k$ is even".

 

[sysprog]

Some text is appropriate, provided it is typeset as text. If I don't know in advance that $n$ is even. I would prefer $\displaystyle\sum_{0 \leq k \leq n,\atop\text{k even}} f(k)$ produced with

\displaystyle\sum_{0 \leq k \leq n,\atop\text{$k$ even}} f(k)

rather than $\displaystyle \sum_{k=0}^{\lfloor n/2 \rfloor} f(2k)$ or any other notation which means "$k$ is even".

I find that to be rather jarringly inconsistent with normal conventions of mathematical expression $-$ I've seen more use of 'where . . . is . . .' in the immediately proximate text rather than text in the expression. I would anticipate seeing a variable or a mathematical subexpression in that position rather than an English-language descriptor. How 'simple' does the 'property' have to be? This seems to me like egregious notational abuse. Would you write $\displaystyle\sum_{0 \leq k \leq n,\atop\text{k perfect_square}} f(k)$?

 

[Office_Shredder]

Yeah, I would prefer that to be written as $\displaystyle \sum_{k\in P(n)} f(k)$ where $P(n)$ is the set of perfect squares less than or equal to n. If you only use this once in a paper, it's wordier, but you're going to hate yourself for not simplifying the notation by the third or fourth time you write the sum.

 

[mathwonk]

apologies for the flip remark, but it may not matter much, as in my experience most people do not understand sigma notation anyway, even when it is both correct and succinct. Hence after some years teaching class, if I wished to be understood, I always wrote out whatever I wanted to say, without using it. verbum sapienti (apologies again). If really needed of course, I could easily live with either the solution by mfb or that of fresh_42, but to me personally words are often clearer than symbols.

 

[sysprog]

@mathwonk, it seems to me that for the most part, the professor writes things in symbols and uses words when reading them aloud or explaining them $-$ the words provide perspicuity, and the symbols avoid ambiguity.

 

[Office_Shredder]

apologies for the flip remark, but it may not matter much, as in my experience most people do not understand sigma notation anyway, even when it is both correct and succinct. Hence after some years teaching class, if I wished to be understood, I always wrote out whatever I wanted to say, without using it. verbum sapienti (apologies again). If really needed of course, I could easily live with either the solution by mfb or that of fresh_42, but to me personally words are often clearer than symbols.

I think it depends a lot on what the class is. If you're trying to teach honors analysis, skipping the sigma notation is doing the students a great disservice.

 

[mathwonk]

touche'. But i presume you do not argue that writing something on the board that students do not understand is doing them a service. in that spirit, i agree, and suggest that when writing sigma notation, one should also write out what it means, or else take the very real chance that you are teaching only to about 5% of the audience. but, in all fairness, you may not have encountered the audiences to which i spent my life teaching! At least I hope not!