I am struggling with this mathematical point. I never saw in class the transition from a sum to an integral. I think I have the intuition, but maybe not the general understanding of how things work.
I was wondering how to compute the sum defined by the sequence :
$$ u_{n} = a n + b $$ with $ a,b \in \mathbb R , n \in \mathbb N $. There are many way to compute the sum, but I was interested in computing it with an integral because when we look at this graph (the function floor(x) )
https://www.desmos.com/calculator/9x8wsf8hwy
we see that if we integrate a constant over an interval of length 1, we get the value of the constant, which here would be exactly the value of our sequence !
so $$ \int_n^{n+1} \left \lfloor{x}\right \rfloor = n $$
but I'm quite unsure about my method. And more importantly, I don't really know how to generalize it.
So my question is, could you please help me giving me methods on how you transform a sum. I know about :
https://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula
https://en.wikipedia.org/wiki/Riemann_sum
But those methods don't ring a bell in my mind, yet. If you could please explain how you construct (using an example perhaps?) the integral given the sum. I also don't understand why on wikipedia they are talking about "aproximation" when I was thinking about computing the true value of the sum.
I m looking to do a sort of repertoire of methods.
thank you !
I'm not aware of any simple way to compute your sum as a Riemann integral. However, if you allow to use Riemann-Stieltjes integral, you can compute your sum using integration by parts:
$$\begin{align} \sum_{k=1}^n k &= \sum_{k=1}^n \int_k^{k+1} \lfloor x \rfloor dx = \int_1^{n+1} \lfloor x \rfloor dx \\&= \lim_{\epsilon\to 0+} \int_{1-\epsilon}^{n+1-\epsilon} \lfloor x \rfloor dx = \lim_{\epsilon\to 0+}\left\{\bigg[\lfloor x \rfloor x\bigg]_{1-\epsilon}^{n+1-\epsilon} - \int_{1-\epsilon}^{n+1-\epsilon} x d\lfloor x \rfloor\right\}\\ &= n(n+1) - \sum_{k=1}^n k \end{align} $$ Rearrange the terms will give you the familiar formula:
$$\sum_{k=1}^n k = \frac{n(n+1)}{2}$$
In general, if you have a function $f(x)$ which are continuous at positive integers, the sum of $f(x)$ over positive integers can be rewritten as a Riemann-Stieltjes integral:
$$\sum_{k=1}^n f(k) = \lim_{\epsilon\to 0+}\int_{1-\epsilon}^{n+\epsilon} f(x) d\lfloor x \rfloor$$ Sometimes integration by parts of this will offer you some insight of the asymptotic behavior of the sum. However, it seldom help one to derive an exact value.
If $f(x)$ can be extended to an entire function on the complex plane. A more power tool is Abel-Plana formula. Compare to Euler-Macluarin formula, it is usually easier to nail down the exact value of the finite part of expansion using AP- rather than EM- formula.
Same as another answer, I'm not sure this helps at all ;-p