Common
Monotone convergence theorem
Theorems on the convergence of bounded monotonic sequences
Monotone convergence theorem
In real analysis, the monotone convergence theorem is any of a number of related theorems proving, under certain conditions, the convergence of monotonic sequences, i.e. sequences that are non-increasing, or non-decreasing. In its simplest form, it says that a non-decreasing bounded-above sequence of real numbers a 1 ≤ a 2 ≤ a 3 ≤ . . . ≤ K {\displaystyle a_{1}\leq a_{2}\leq a_{3}\leq ...\leq K} converges to its smallest upper bound, its supremum. Likewise, a non-increasing bounded-below sequence converges to its largest lower bound, its infimum. In particular, infinite sums of non-negative numbers converge to the supremum of the partial sums if and only if the partial sums are bounded. For non-negative...
Open boosters, collect cards and trade them. Free to play.
Play for free