Editing Monotone convergence theorem (section)

===Examples===

====Matrices====

The theorem states that if you have an infinite matrix of non-negative real numbers <math> a_{i,k} \ge 0</math> such that the rows are weakly increasing and each is bounded <math>a_{i,k} \le K_i</math> where the bounds are summable <math>\sum_i K_i <\infty</math> then, for each column, the non decreasing column sums <math>\sum_i a_{i,k} \le \sum K_i </math> are bounded hence convergent, and the limit of the column sums is equal to the sum of the "limit column" <math> \sup_k a_{i,k}</math> which element wise is the supremum over the row.

====''e''====

Consider the expansion
:<math> \left( 1+ \frac1k\right)^k = \sum_{i=0}^k \binom ki \frac1{k^i} 
</math>
Now set 
:<math> a_{i,k} = \binom ki \frac1{k^i} = \frac1{i!} \cdot \frac kk \cdot \frac{k-1}k\cdot \cdots \frac{k-i+1}k
</math>
for <math> i \le k </math> and <math> a_{i,k} = 0</math> for <math> i > k </math>, then <math>0\le a_{i,k} \le a_{i,k+1}</math> with <math>\sup_k a_{i,k} = \frac 1{i!}<\infty </math> and 
:<math>\left( 1+ \frac1k\right)^k  = \sum_{i =0}^\infty a_{i,k}</math>.   
The right hand side is a non decreasing sequence in <math>k</math>, therefore
:<math> \lim_{k \to \infty}
\left( 1+ \frac1k\right)^k = \sup_k \sum_{i=0}^\infty a_{i,k} = \sum_{i = 0}^\infty \sup_k  a_{i,k} = \sum_{i = 0}^\infty \frac1{i!} = e</math>.