Riemann sums.

Today, I studied Riemann sums as preliminary to fully understand intuitively the concept of integrals.

During the study session, I decided to get the limit value of the Riemann sum of $ f(x) = x^2 $ in the interval [1,3] but I was doing a rookie mistake during the process.

$$ \sum_{k=1}^{n} c \cdot k = c \sum_{k=1}^{n} k ,\ \text{NOT} ,\ n \cdot c \sum_{k=1}^{n} k $$

For summation of constant values:

$$ \sum_{k=1}^{n} c = n \cdot c $$

Shifting the index in a summation

For example:

$$ \sum_{i=0}^{n-1} i $$

We make $ k = i + 1 $

$$ \sum_{k=1}^{n} (k - 1) $$