The Monotone Convergence Sequence

The Monotone Convergence Criterion

The monotone sequence is one important aspect of sequences .
A monotone sequence is any sequence ${s_n}$ that is either increasing,decreasing or constant. Any sequence can be verified if it is monotonic using the formula, $\text{Monotone formula}=s_{n+1}-s_n$.

Now A sequence $s_n$ of $\mathbb{R}$ is called the following:
(i) Monotone non-decreasing(Increasing) if $s_{n+1}\geq{s_n}$, for all $n\in\mathbb{N}$.
(ii) Strictly Monotone Increasing: if $s_{n+1}>s_n$ for all $n\in\mathbb{N}$.
(iii) Monotone non-decreasing(decreasing) if $s_{n+1}\leq{s_n}$ for all $n\in\mathbb{N}$.
(iv) strictly Monotone Decreasing if $s_{n+1}<s_n$ for all $n\in\mathbb{N}$.

Fundamental Theorem of Monotone Sequence
(i) A monotone increasing sequence of real number which is bounded above converges.
(ii) A monotone decreasing sequence of real number which is bounded below converges.
(iii) Any sequence that is bounded and is monotonic is convergent.

from all this we can observe that the monotonic sequences are easier to deal with than the sequences that can go up and down.
In my next topic i will be treating how to carry out evaluation(solutions) of this monotone sequences.

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