To solve the series:

$\frac{1}{5 \times 9} + \frac{1}{9 \times 13} + \frac{1}{13 \times 17} + \cdots + \frac{1}{101 \times 105}$

I would use Partial Fraction Decomposition combined with Summation of Telescoping Series as the primary techniques. Here's a breakdown of the approach:

1. Partial Fraction Decomposition:

Each term can be split into simpler fractions using partial fractions. For instance, the general term $\frac{1}{n(n+4)}$ can be rewritten as:

$\frac{1}{n(n+4)} = \frac{A}{n} + \frac{B}{n+4}$

Solving for $A$ and $B$, we find that:

$\frac{1}{n(n+4)} = \frac{1}{4} \left(\frac{1}{n} - \frac{1}{n+4}\right)$

2. Telescoping Series:

Once each term is decomposed, the series forms a telescoping series, where many terms cancel out. For example, after decomposing each term, you might have something like:

$\frac{1}{4} \left( \left(\frac{1}{5} - \frac{1}{9}\right) + \left(\frac{1}{9} - \frac{1}{13}\right) + \cdots + \left(\frac{1}{101} - \frac{1}{105}\right) \right)$

In this structure, all intermediate terms cancel out, and you are left with the first and last terms of the series.

3. Summing the Remaining Terms:

After cancellation, you'll only need to sum the remaining terms, which are:

$\frac{1}{4} \left( \frac{1}{5} - \frac{1}{105} \right)$

This simplifies the calculation to:

$\frac{1}{4} \left( \frac{100}{525} \right) = \frac{25}{525} = \frac{1}{21}$

Thus, the sum of the series is $\frac{1}{21}$.

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Relative Questions:

1. How does the telescoping property help simplify series summation?
2. Can partial fraction decomposition be used for any type of rational function?
3. What is the difference between telescoping series and geometric series?
4. How can you verify the final result of a telescoping series using limits?
5. In which other types of problems can partial fraction decomposition be applied?

Tip:

In any series problem, look for patterns that simplify through cancellation, such as telescoping effects. It drastically reduces the complexity of the final summation.