Notation question and help with $\Sigma_k \binom{n}{2k}$

Question

What does $\Sigma_k$ mean? This is on the review for the test, but I've literally never seen it before. Am I just to assume that it's the same as $\sum_{k=0}^n$?

How do I simplify the sum of a binomial coefficient? Are they trying to get me to say its the same as $2 \sum_{k=0}^{\frac{n}{2}} \binom{n}{2k}$ ?

Yes, you can just treat it as the sum from $k=0$ to $\mathrm{floor}(n/2)$, since the way binomial coefficients work, all other terms are treated as $0$. When you see an expression like that, think of it as the sum over all terms where the expression makes sense. — Kevin Long, Sep 20 '17 at 18:38
Yes, $\sum_k$ is sometimes used as a (messy and imprecise) shorthand to mean that $k$ is the index variable and you sum over all possible values of $k$ (which here likely implies we are treating $k$ as an integer, and the only integer values of $k$ which result in a nonzero summand are in the range $[0,n/2]$) — JMoravitz, Sep 20 '17 at 18:40
@Dr. Sonnhard Graubner That's a big name for $2^{n-1}$ (for $n>0$). — , Sep 20 '17 at 18:43
Thanks a lot everybody! I see now the answer is $2^{n-1}$, much appreciated. — XRBtoTheMOON, Sep 20 '17 at 19:02

gen-ℤ ready to perish · Accepted Answer · 2017-09-27T20:53:49.110

$$\sum_k\cdots$$

is read “the sum over all $k$ of …”, “the sum indexed by $k$ of …,” or something similar.

It is used when the limits of $k$ are unknown, implied, or inconvenient to write, but when it’s also still necessary to indicate the index of the sum.

For example, the $w$-weighted average of data $x$ is $$\sum_kw_kx_k$$ where $w_k$ is the percentage weight of the $k$th datum. But, in general, you never know exactly how many $k$s there are.

Sometimes, you’ll see a sum with no index, such as $$\sum\vec F$$ because it’s unequivocal what is varying across the sum. Above, $\vec F$ is varying. In the other cases, $k$ was definitely varying, which would in turn possibly change the values of $w$ and $x$, but possibly not.

Notation question and help with $\Sigma_k \binom{n}{2k}$

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