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The problem asks to examine the convergence of the following series:

\[
\sum_{n=1}^{\infty} \frac{1}{n(n+1)}
\]

Let's break down and solve this series step by step.

### 1. **Decomposing the General Term:**

We start with the general term of the series:

\[
\frac{1}{n(n+1)}
\]

We can simplify this expression by performing **partial fraction decomposition**:

\[
\frac{1}{n(n+1)} = \frac{A}{n} + \frac{B}{n+1}
\]

Multiplying both sides by \(n(n+1)\) to clear the denominator:

\[
1 = A(n+1) + Bn
\]

Expanding the right-hand side:

\[
1 = A n + A + B n
\]

Group the terms involving \(n\):

\[
1 = (A + B)n + A
\]

For this equation to hold true for all \(n\), we must have:

\[
A + B = 0 \quad \text{and} \quad A = 1
\]

Thus, \(A = 1\) and \(B = -1\). Therefore, the partial fraction decomposition is:

\[
\frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1}
\]

### 2. **Writing the Series:**

Now, substituting this into the sum, we have:

\[
\sum_{n=1}^{\infty} \frac{1}{n(n+1)} = \sum_{n=1}^{\infty} \left(\frac{1}{n} - \frac{1}{n+1}\right)
\]

This is a **telescoping series**, where most terms cancel out:

\[
\left( \frac{1}{1} - \frac{1}{2} \right) + \left( \frac{1}{2} - \frac{1}{3} \right) + \left( \frac{1}{3} - \frac{1}{4} \right) + \cdots
\]

### 3. **Finding the Sum:**

In a telescoping series, all intermediate terms cancel out, leaving:

\[
1 - \lim_{n \to \infty} \frac{1}{n+1} = 1 - 0 = 1
\]

### 4. **Conclusion:**

The series converges, and its sum is **1**.

Would you like more details, or do you have any questions?

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Here are some related questions for further understanding:
1. How does partial fraction decomposition help in summing telescoping series?
2. What other techniques can be used to check for convergence of series?
3. How do you apply the ratio test or root test for series convergence?
4. Can all series with rational terms be decomposed into partial fractions?
5. How does the behavior of the harmonic series compare to this one?

**Tip:** When dealing with series, always consider if a telescoping pattern appears, as it can significantly simplify the process of summing the series.

Examine the convergence of the following series: 1/1.2 + 1/2.3 + 1/3.4 + 1/4.5 + ...

Learn how to examine the convergence of the series 1/1.2 + 1/2.3 + 1/3.4 + 1/4.5 + ... using partial fraction decomposition and telescoping series. Suitable for undergraduate-level mathematics.

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