OEF Sequences --- Introduction ---

This module actually contains 16 exercises on infinite sequences: convergence, limit, recursive sequences, ...

Two limits

Let () be an infinite sequence of real numbers. If one has

and for ,

what can be said about its convergence? (You should choose the most pertinent consequence.)

Comparison of sequences

Let () and () be two sequences of real numbers where () converges towards . If one has

,

what can be said about the convergence of ()? (You must choose the most pertinent consequence.)

Growth and bound

Let () be a sequence of real numbers. If () is , what can be said about its convergence (after its existence)?

Convergence and difference of terms

Let be a sequence of real numbers. Among the following assertions, which are true, which are false?
  1. If , then .

  2. If , then .

Convergence and ratio of terms

Let be a sequence of real numbers. Among the following assertions, which are true, which are false?
  1. If , then .

  2. If , then .

Epsilon

Let be a sequence of real numbers. What does the condition

imply on the convergence of ? (You must choose the most pertinent consequence.)

Fraction 2 terms

Compute the limit of the sequence (un), where

Fraction 3 terms

Compute the limit of the sequence (un), where

Fraction 3 terms II

Compute the limit of the sequence (un), where

WARNING IN this exercise, approximative replies will be considered as false! Type pi instead of 3.14159265, for example.

Growth comparison

What is the nature of the sequence (un), where

 ?

Monotony I

Study the growth, sup, inf, min, max of the sequence (un) for n ge , where

.

Write for a value that does not exist, and or - for +infty or -infty.

Monotony II

Study the growth, sup, inf, min, max of the sequence (un) for n ge , where

.

Write for a value that does not exist, and or - for +infty or -infty.

Powers I

Compute the limit of the sequence (un), where

Powers II

Compute the limit of the sequence (un), where

Type no if the sequence is divergent.

Recursive function

The sequence such that
is a recursive sequence defined by for a certain function . Find this function.

Recursive limit

Find the limit of the recursive sequence such that
The most recent version
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