Everybody Hello, I'm confused about the following integral:
Consider the following:
Riemann Integral: $\int f dx:=\{\sum_{E\in\mathcal{E}} f(E)\lambda(E)\}_\mathcal{E}$
Domain: $I:=[0,1]$
Function: $f(x):=\frac{1}{\sqrt x}$, $f(0):=0$
The Problem is now:
This should be Riemann integrable on the compact unit interval as we know by formal antiderivative and neglecting the boundary, however, comparing upper and lower sum or equivalently tagged partitions it shouldn't be Riemann integrable ...did I miss sth?
That is in Detail:
Take the refining sequence of partitions: $\mathcal{E}_n:=\{(\frac{k-1}{n},\frac{k}{n}]\}_{k=1}^n$
Then compute the upper sum by taking the suprema: $\sum_{k=1}^n\sup f(E_k)\lambda(E_k)=\infty\frac{1}{n}+\sqrt{\frac{n}{1}}\frac{1}{n}+\sqrt{\frac{n}{2}}\frac{1}{n}\ldots=\infty$
...this already tells us it shouldn't be Riemann integrable
Moreover, It seems as if the function even fails to be integrable as improper integral.
Am I missing sth when simply considering equidistant partitions? Doing so I end up with sum variaty of the Riemann Zeta Function being divergent...