What is the expanded form of $\sum_\limits{0}^{0}{f(x)}$?

Question

The is a MCQ in my math book which says the following:

Expanded form of $\sum\limits_0^0{f(x)}$ is:

1) $0$

2) $f(0)$

3) $1$

4) None

I don't know which one is correct but one of the first two is correct.

Answer 1

Written correctly the formula should be:

$$\sum_\limits{x=0}^0 f(x)$$

This means $x$ ranges between $0$ and $0$ and so only takes the value $0$.

Therefore the answer is b, $f(0)$.

$x$ is not often used an the index value of a summation, $i,j,k,n$ are much more commonly used instead. $x$ is usually considered to be a real variable, and so the question is slightly misleading.

Answer 2

The correct answer is $f(0)$ (according to Wolfram|Alpha). The reason is if the bounds are the same, and you are computing a summation of a function, the answer will be the function with the bounds passed as the value. Therefore, the answer is $f(0)$ for any expression defined as $f(x)$. The only case in which this "isn't true" is for $x = \infty$. Since uses of infinity in regular expresssions evaluate to $\infty$, the answer here is $\infty$. But still, the answer is: $$\sum_{x = 0}^0 f(x) = f(0)$$