This may look as a stupid question but it will really help me a lot in understanding some things.
Suppose I take the simple function $f(x) = x$. Now I will evaluate two different operations:
$$\int_0^x x'\ \text{d}x' = \frac{x^2}{2}$$
$$\sum_{x' = 0}^x x' = \frac{x\cdot(x+1)}{2} = \frac{x^2}{2} + \frac{x}{2}$$
Now: the integral from $0$ to $x$ is clearly understood. Indeed the result is nothing but the area of a square of size $x$, divided by two. Indeed the area under that curve is nothing but the area of a triangle which is the perfect half of a square.
Now the sum. What does the sum means? The result may be interpreted as the area of a triangle of sides $x$ and $x+1$ but.. why?
If the interpretation of being an area is correct, why it's not the same? Is it because the series travels only on integers, whilst the integral is the continue? Or the area interpretation is totally wrong.
I have to apologize if sometimes I ask for dumb questions, but sometimes doubts arise like so..