Numerical sequences

Direction of variation of a sequence

Definition

  • A sequence (u_{n})_{n\in 
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
} is increasing if for all n\geq 0: u_{n+1}\geq u_{n}..

  • A sequence (u_{n})_{n\in 
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
}is decreasing if for all n\geq 0:u_{n+1}\leq u_{n}.

  • A sequence (u_{n})_{n\in 
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
} is constant if for all n\geq 0:u_{n+1}=u_{n}.

Example

Study the direction of variation of the sequence u defined on \mathbb{N} by u_{n}=4n+3

The answer: u_{n+1}=4(n+1)+3=4n+4+3=4n+7

for all n\geq 0 we have u_{n+1}-u_{n}=(4n+7)-(4n+3)=4n+7-4n-3=4>0. Then the sequence (u_{n})_{n\in 
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
} is increasing

Proposition

Let f be a function defined on the interval [0;+\infty \lbrack and, for any natural number n, u_{n}=f(n).

  • If the function f is increasing on [0;+\infty \lbrack, then the sequence u is increasing.

  • If the function f is decreasing on the interval [0;+\infty \lbrack, then the sequence u is decreasing.

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