Notes - Analysis III TT23, Riemann sums
Flashcards
What is the $\text{mesh}$ of a partition $\mathcal P$?
The length of the smallest subinterval in the partition.
Draw a picture showing the difference between the Darboux formulation of integration and the Riemann formulation of integration?
Two pictures,
- Darboux: step functions above/below a function, with $I(\phi) = \sum c _ i (x _ {i-1} - x _ i)$
- Riemann: tagged partition, with $\Sigma(f; \mathcal P^{(i)}, \vec \xi^{(i)}) = \sum f(\vec \xi^{(i)} _ j) (x _ {j-1} - x _ j)$
What is
\[\Sigma(f; \mathcal P^{(i)}, \vec \xi^{(i)})\]
?
where $\vec \xi = (\xi _ 1, \ldots, \xi _ n)$ and $\xi _ j \in [x _ {j-1}, x _ j]$.
What theorem relates Riemann sums and Darboux integrals?
Let $f: [a, b] \to \mathbb R$ be a function. Let $\mathcal P^{(i)}$ be a sequence of partitions with $\text{mesh}(\mathcal P^{(i)}) \to 0$. Then $f$ is integrable with value $c$ in the Darboux sense if and only if
\[\lim_{i \to \infty} \Sigma(f, \mathcal P^{(i)}, \vec \xi^{(i)}) = c\]for any choice of $\vec \xi^{(i)}$.
What is true about Riemann sum and step functions?
Every Riemann sum is equivalent to the integral of some step function.