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Partial sums are the sum total of the first 'n' terms of a given series. When evaluating a series, it's essential to track these partial sums to determine if the series converges or diverges. The sequence of partial sums is crucial as it gives insight into the behavior of the entire series.
For instance, if you have a series such as \( \sum_{n=2}^{\infty} a_n \), the partial sums are represented as \( S_n = a_2 + a_3 + ... + a_n \).
Analyzing these sums, especially plotting them can visually hint at whether they approach a certain value or not.

• If the sequence of partial sums approaches a finite number as 'n' increases, the series converges.
• If the partial sums increase indefinitely, the series diverges.

The exercise given involves finding at least 10 partial sums to visualize and determine the convergence or divergence of a series. Graphing these sums alongside the terms themselves often provides a clearer understanding of the series' behavior over time.

A telescoping series is a special type of series where successive terms cancel each other out. This property often simplifies the computation of its partial sums. The original series \( \sum_{n=2}^{\infty} \left( \frac{1/2}{n} - \frac{1/2}{n+2} \right) \) exemplifies this trait.
A telescoping series typically involves two parts that, when added, cancel out intermediate terms of prior partial sums. You will often see a pattern forming where terms like "\( \frac{1/2}{k} \)" in one partial sum cancel with "\( -\frac{1/2}{k} \)" in another.
This cancellation process makes results easier to compute and offers insight into whether the entire series converges. Telescoping series are beneficial for calculating particular sums and recognizing convergence, as parts of the series quickly reduce to simpler forms.

• Identify the form \( a_n - a_{n+d} \) where terms linearly reduce.
• Calculate partial sums more smoothly by focusing on the few remaining terms after cancellation.

In the exercise, after writing the series in telescoping form, you observe that the series doesn't condense enough and the partial sums grow, pointing towards divergence.

Partial fraction decomposition is a method used to break down complex rational expressions into simpler fractions that are easier to work with. It's particularly useful when dealing with algebraic fractions like \( \frac{1}{n(n+2)} \).
The process involves expressing the given fraction in terms of sums of simpler fractions. For the series in the exercise, performing partial fraction decomposition transforms \( \frac{1}{n(n+2)} \) into \( \frac{A}{n} + \frac{B}{n+2} \).
Knowing this allows us to solve for constants \( A \) and \( B \). Here, setting the equation \( 1 = A(n+2) + Bn \) and solving gives \( A = \frac{1}{2} \) and \( B = -\frac{1}{2} \). Therefore, the fraction becomes \( \frac{1/2}{n} - \frac{1/2}{n+2} \).
Partial fraction decomposition provides a clearer structure, enabling the further exploration and simplification of the series.

• Used mainly for rational expressions where direct summation isn’t trivial.
• By decomposing, focus can be shifted to easier manipulations like implementing telescoping series.

In this problem, decomposition plays a critical role by turning a single, challenging expression into pieces that quickly lead into a telescoping series, crucial for estimating behavior of the series.