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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..

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  • $\begingroup$ Concerning your parallel between integral and sum see a generalization using Pochhammer symbol here (this may thus be generalized to any polynomial at least...). Umbral Calculus may be of interest for further generalizations. $\endgroup$ Mar 22, 2016 at 12:51

2 Answers 2

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The sum is the area of a bunch of rectangles of width $1$ situated at $x=0,1,\ldots$ with height given by your integration function.

If you keep the total region of integration constant, but increase the amount of rectangles (decreasing their width simultaneously to account for the same total interval length), you will get the integral in the limit under some nice conditions. This is the idea behind the standard Riemann integration.

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For the sum (you should fix the summation variable), consider the following picture:

enter image description here

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