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I want the resolution of the last question to demonstratthat this serie could be write as a fraction of impair sur pair

To resolve the last question and demonstrate how the given series can be written as a fraction of odd over even numbers, let's start by analyzing the problem from the image.

### Problem Summary
The problem asks to consider the series:
\[
S = \sum_{n=1}^{\infty} \frac{1}{n(n+2)}
\]
and show that it can be written as a fraction involving odd and even numbers.

### Step 1: Simplify the Series
First, let's express the general term \(\frac{1}{n(n+2)}\) using partial fraction decomposition:
\[
\frac{1}{n(n+2)} = \frac{A}{n} + \frac{B}{n+2}
\]
To find \(A\) and \(B\), solve the following equation:
\[
1 = A(n + 2) + Bn
\]
Expanding and matching coefficients:
\[
1 = An + 2A + Bn
\]
\[
1 = (A + B)n + 2A
\]
From this, we equate coefficients of \(n\) and the constant term:
\[
A + B = 0 \quad \text{and} \quad 2A = 1
\]
Solving these, we get \(A = \frac{1}{2}\) and \(B = -\frac{1}{2}\). Therefore:
\[
\frac{1}{n(n+2)} = \frac{1}{2} \left(\frac{1}{n} - \frac{1}{n+2}\right)
\]

### Step 2: Express the Series
Now the series can be rewritten as:
\[
S = \sum_{n=1}^{\infty} \frac{1}{2} \left(\frac{1}{n} - \frac{1}{n+2}\right)
\]
This is a telescoping series, which simplifies as:
\[
S = \frac{1}{2} \left(1 + \frac{1}{2} + \frac{1}{3} - \frac{1}{3} - \frac{1}{4} - \frac{1}{5} + \frac{1}{5} + \frac{1}{6} + \dots\right)
\]
Most of the terms cancel out, leaving:
\[
S = \frac{1}{2} \left(1 + \frac{1}{2}\right)
\]
So:
\[
S = \frac{1}{2} \times \frac{3}{2} = \frac{3}{4}
\]

### Final Expression
The sum of the series can thus be written as \(\frac{3}{4}\), which is a fraction of an odd number over an even number. This concludes the demonstration.

Do you need further details or have any questions?

### Related Questions
1. How would the series change if it started from \(n=2\) instead of \(n=1\)?
2. Can you generalize the partial fraction decomposition for a series \(\frac{1}{n(n+k)}\) where \(k\) is any integer?
3. What is the convergence condition for the series \(\sum_{n=1}^{\infty} \frac{1}{n(n+c)}\) for different values of \(c\)?
4. How can we interpret the result \(\frac{3}{4}\) geometrically or in terms of probability?
5. What are other methods to evaluate this type of series besides partial fraction decomposition?

### Tip
When working with infinite series, always check if it can be simplified using techniques like telescoping, which often makes the problem much easier to solve.

I want the resolution of the last question to demonstrate that this series could be written as a fraction of odd over even numbers.

Explore how to resolve a mathematical series using partial fraction decomposition and telescoping series methods to demonstrate that it can be expressed as a fraction of odd over even numbers. Detailed solution with step-by-step breakdown suitable for undergraduate mathematics.

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