Write the following series in the abbreviated \(\sum\) form. $$ \frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\frac{1}{4 \cdot 5}+\frac{1}{5 \cdot 6}+\cdots $$

Summation notation is a compact way to represent the sum of a series of terms. It's often denoted by the Greek letter Sigma (\sum). Summation notation allows us to easily express and work with long sums in a concise manner. Let's break down the notation used in the exercise:

• The symbol \(\sum\) indicates that we are summing a sequence of terms.
• The subscript (\(n=2\) here) tells us the starting index of the summation.
• The general term inside the summation (\(\frac{1}{n(n+1)}\)) describes each term that needs to be summed, based on the value of \(n\).
• The superscript (\(\text{∞}\)) indicates that the series goes on indefinitely.

When you see the summation notation, it is essential to understand both the starting index and the formula for the general term. This will help you expand the series if necessary.

Pattern recognition in a series is the process of identifying a systematic relationship between the terms. Recognizing the pattern helps in writing and working with the series more efficiently. In our exercise, notice that each term can be written in the form \(\frac{1}{n(n+1)}\). This is a key observation.
Understanding the pattern allows you to write out the series without manually calculating each term. For instance, the given series:
\(\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\frac{1}{4 \cdot 5}+\frac{1}{5 \cdot 6}+\ldots\)
can be recognized as terms of the form \(\frac{1}{n(n+1)}\) starting from \(n=2\) and continuing indefinitely. Identifying patterns is a fundamental skill in mathematics as it simplifies complex problems and aids in deeper understanding.

A mathematical series is the sum of the terms of a sequence. Sequences themselves are ordered lists of numbers which often follow a specific rule or pattern. In our example,
the sequence \(2 \cdot 3, 3 \cdot 4, 4 \cdot 5,...\),
is what generates the terms of the series. When we sum the reciprocals of these products, we form our series:
\(\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\frac{1}{4 \cdot 5}+\frac{1}{5 \cdot 6}+\ldots\).
Some important points to remember about series:

• A series can be finite or infinite.
• The general term dictates how each value in the series is formulated.
• Understanding whether a series converges (sums to a specific value) or diverges (grows without bound) is crucial in advanced mathematics.

In our case, the series: \[\text{{\sum}}_{{n=2}}^{{\infty}}\frac{1}{n(n+1)}\]
is an infinite series that we defined using the summation notation.