Analysis Basics

Take 15 minutes to prepare this exercise.

Then, if you lack ideas to begin, look at the given clue and start searching for the solution.

A detailed solution is then proposed to you.

If you have more questions, feel to ask then on the forum.

Question

We denote by a strictly positive real number and by a real number belonging to interval

We define two real sequences and by: and for any integer greater or equal to

and

Find the limits of sequences and

We can set and look for the expressions of the sequences depending on

Indice

Express and in the form of a product of

To find the limit of write it as quotient of starting from

Solution

Let us assume:

The expressions of and are therefore justified by recursion for any natural integer

There remains to calculate the limit when goes to infinity of

We obtain recursively:

We infer:

is equivalent to when goes to infinity, therefore: goes towards when goes to infinity. This implies that both sequences converge towards .

PrécédentPrécédentSuivantSuivant
AccueilAccueilImprimerImprimerRéalisé avec Scenari (nouvelle fenêtre)