# 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 , where
.

### Fraction 3 terms

Compute the limit of the sequence (un), where

### Fraction 3 terms II

Compute the limit of the sequence , 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 for , where
.
Write for a value that does not exist, and or - for + or -.

### Monotony II

Study the growth, sup, inf, min, max of the sequence for , where
.
Write for a value that does not exist, and or - for + or -.

### Powers I

Compute the limit of the sequence , 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

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