# What does it mean for a sequence to be monotone?

It means that the sequence is always either increasing or decreasing, it the terms of the sequence are getting either bigger or smaller all the time, for all values bigger than or smaller than a certain value.

Here is the precise definitions :

Furthermore, there is a theorem which states that every bounded, momotonic sequence is convergent.

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A sequence is said to be monotone if it consistently either increases or decreases as you move along the sequence. Specifically, if each term in the sequence is greater than or equal to the preceding term, the sequence is called monotone non-decreasing or simply monotone increasing. Conversely, if each term is less than or equal to the preceding term, the sequence is termed monotone non-increasing or monotone decreasing. In both cases, the direction of change remains the same throughout the sequence.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- What is the result of lim x → ∞ (-x-1)/((x-1)(x-Sqrt(x^2+x+1)) ?
- Show that lim x->a (x^3/8-a^3/8)/(x^5/3-a^5/3)?
- How do you solve the series (7 - 9 + (81/7) - (729/49) + ... )?
- How do you use the ratio test to test the convergence of the series #sum_(n=1)^oo (n!)/((2n+1)!)#?
- How do you determine if the improper integral converges or diverges #int ln(sin(x))# from 0 to pi/2?

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