I have function $$ f(x) = \frac{1}{x^2} $$ I want to riemann (I don't know whether what I mean is actually Riemann or Darboux integration) integrate it on the interval $$ x \in \left[2,3\right] $$
What I could do is partition the interval into subintervals, first stating that $$ 2 = x_0 < x_1 < x_2 < \dots < x_{n-1} < x_n =3 \implies P=(x_0, \: \dots \:, x_n)$$
And
$$ m_i = \inf_{x \in [x_{i-1}-x_{i}]}{\left( f(x) \right)} $$ $$ M_i = \sup_{x \in [x_{i-1}-x_{i}]}{\left( f(x) \right)} $$
So I can tell the darboux (??) sums:
Lower: $L_{f,P} = \sum_{i=1}^{n}{(x_i-x_{i-1})m_i}$
Upper: $U_{f,P} = \sum_{i=1}^{n}{(x_i-x_{i-1})M_i}$
I could find $m_i$ and then, for example, $L_{f,P}$ is:
$$L_{f,P} = \sum_{i=1}^{n}{(x_i-x_{i-1}) \left( \frac{1}{x_i^2} \right)}$$
How do I continue?
I know that I should find what this value approaches as $n \to \infty$ and check whether $U_{f,P}$ approaches the same number to find the integral. But I can't take limits of sums and I don't know how I should simplify that...
Please, if possible, I'd like simple and beginner-level answers
Let $I \subset\mathbb{R}$ be a closed interval and $f:I\to\mathbb{R}$ be a bounded function. Let \begin{eqnarray} \mathrm{L}f := \sup_{P \text{ is a partition of }I}L_{f,P}\\ \mathrm{U}f := \inf_{P \text{ is a partition of }I}U_{f,P}. \end{eqnarray} and $|P| > 0$ be the maximum length among the subinterevals in $P$, where $P$ is a partiton of $I$.
Then there is a theorem that
for all $\varepsilon > 0$ there exists $\delta > 0$ s.t. if a partition $P$ of $I$ satisfies $|P| < \delta$ then \begin{eqnarray} \left| L_{f,P}-Lf \right| < \varepsilon\\ \left| U_{f,P}-Uf \right| < \varepsilon. \end{eqnarray}
Hence you can calculate the lower and upper Riemannian integral with partitions whose subintervals are of length $\frac{1}{n}$.
In order to calculate the lower one, calculate \begin{eqnarray} \frac{1}{n}\sum_{i=1}^n\frac{1}{\left( 2 + \frac{i}{n} \right)\left( 2 + \frac{i-1}{n} \right)} \end{eqnarray} instead of \begin{eqnarray} \frac{1}{n}\sum_{i=1}^n\frac{1}{\left( 2 + \frac{i}{n} \right)^2} \end{eqnarray} and then estimate the difference to each other.