What is $\sum_ {i=1}^{10}2$? What is $\displaystyle\sum\limits_{i=1}^{10}2$?
Is it  $$\sum\limits_{i=1}^{10}2  = 200$$
or
$$\sum\limits_{i=1}^{10}2 = 2$$
or
$$\sum\limits_{i=1}^{10}2= 0$$
or none of the above? 
I feel a little ashamed to even ask such a basic question, but I have looked over the definition of $\sum$'s and I can't wrap my head around it. The question comes from pg 40 of Apostle's Calculus which I am independently studying. 
 A: $$ \sum_{i=1}^{10}2=2+2+2+2+2+2+2+2+2+2=20$$
A: The summation sign written as $$\sum_{k = a}^m f(k) $$
is defined such that you take values starting at $a$ counting all the way up to $m$ and sub them in for $k$ in $f(k)$. Each one of these separate values are added together. 
In your case, there is no such $k$ in the $f(k)$ as $f$ is the constant $f(k) = 2$. You count from $1$ through to $10$, which means you will be adding $f(1)$ through to $f(10)$ like so $$f(1) + f(2) + \ldots + f(10).$$
As $f(k) = 2$, this simplifies to $$\overset{\text{10 times}}{2 + 2 + \ldots + 2} = 10(2) = 20.$$
A: The argument of the sum is a function of the iterator $i$; in this case it is the constant function 2.
When dealing with sums you can always think them as
$$ \sum_{i=1}^{10} f(i) = f(1) + f(2) + \dots + f(10) $$
In this case we have $f(1)=f(2)= \dots = f(10) = 2$ and so
$$ \sum_{i=1}^{10} f(i) = 2 + 2 + \dots + 2 = 2 \cdot 10 = 20. $$
A: You can think of it this way: let $a_1,\dots a_n$ be real numbers, then we define $$\sum\limits_{k=1}^n a_k:=a_1+a_2+a_3+\ldots +a_n.$$ In your case we have $a_1=2,a_2=2,a_3=2,\ldots,a_{10}=2$, thus $$\sum\limits_{i=1}^{10}2=\underbrace{2}_{=a_1}+\underbrace{2}_{=a_2}+2+2+2+2+2+2+2+\underbrace{2}_{=a_{10}}=20.$$
