Question

For each of the following series, use the sequence of partial sums to determine whether the series converges or diverges. $$ \sum_{n=1}^{\infty} \frac{1}{(n+1)(n+2)} $$

Short answer

The series converges to \( \frac{1}{2} \).

Step-by-step solution

Rewrite the General Term

The general term of the series is given by \( a_n = \frac{1}{(n+1)(n+2)} \). Notice this can be rewritten using partial fraction decomposition. We can express \( a_n \) as a difference of two fractions:\[ a_n = \frac{1}{n+1} - \frac{1}{n+2} \]

Write the Sequence of Partial Sums

Using the decomposition from Step 1, the partial sum \( S_N \) can be expressed as the sum of these fractions for all \( n \) from 1 to \( N \):\[S_N = \sum_{n=1}^{N} \left( \frac{1}{n+1} - \frac{1}{n+2} \right)\]

Simplify the Partial Sum

Notice that the partial sums form a telescoping series:- \( S_N = \left( \frac{1}{2} - \frac{1}{3} \right) + \left( \frac{1}{3} - \frac{1}{4} \right) + \ldots + \left( \frac{1}{N+1} - \frac{1}{N+2} \right) \).Many terms cancel out, leaving:\[S_N = \frac{1}{2} - \frac{1}{N+2}\]

Determine Convergence/Divergence

Consider the behavior of \( S_N = \frac{1}{2} - \frac{1}{N+2} \) as \( N \to \infty \):- As \( N \) increases, \( \frac{1}{N+2} \to 0 \).- This implies that \( S_N \to \frac{1}{2} \).Since the sequence of partial sums \( S_N \) converges to \( \frac{1}{2} \), the original series converges.

Key concepts

Partial Sums

When you look at a series, one important tool to understand its behavior is the use of partial sums. A partial sum is created by taking the sum of the first few terms of the series.
For instance, if you have a series written as \( \sum_{n=1}^{\infty} a_n \), the partial sum \( S_N \) would be \( a_1 + a_2 + \ldots + a_N \).
The sequence of partial sums allows us to examine what happens as we add more terms. This is crucial in determining if a series converges, meaning it approaches a certain number, or diverges, meaning it grows larger without bound.
In the original exercise, the series \( \sum_{n=1}^{\infty} \frac{1}{(n+1)(n+2)} \) is studied by looking at its sequence of partial sums.

Telescoping Series

A telescoping series is a type of series where many terms cancel out when you write the partial sums. This often leads to a much simpler expression for the partial sum compared to the original series.
The concept comes from the idea of a telescope, where what you see at the beginning fades away, focusing on what's new. When dealing with telescoping series, you find that only a few terms from the start and a few from the end remain.
For our series, the partial sums simplify significantly because they are telescoping:

• \( S_N = (\frac{1}{2} - \frac{1}{3}) + (\frac{1}{3} - \frac{1}{4}) + \ldots + (\frac{1}{N+1} - \frac{1}{N+2}) \)
• Terms like \( \frac{1}{3} \), \( \frac{1}{4} \), and so on, neatly cancel with later terms.
• This leaves us with a very simple expression: \( S_N = \frac{1}{2} - \frac{1}{N+2} \)

This ability to cancel many terms efficiently helps in easily determining if a series converges or not.

Partial Fraction Decomposition

Partial fraction decomposition is a strategy used to rewrite complex fractions into simpler components.
It is especially helpful for dealing with rational expressions, like the one in our series.
In the original problem, we have the term \( \frac{1}{(n+1)(n+2)} \). Using partial fraction decomposition, this is rewritten as:

• \( a_n = \frac{1}{n+1} - \frac{1}{n+2} \)
• This new representation shows the terms are made up of simple fractions, making them easier to work with in a series context.

This approach allows for easier simplification and evaluation of the series, as it reveals the telescoping nature of the series.

Infinite Series

An infinite series is the sum of an infinite sequence of numbers. The key question is whether the infinite series converges, that is, whether it approaches a specific finite value as you add more and more terms.
To evaluate if an infinite series converges, we often look at the sequence of its partial sums.
In the case of the series provided, \( \sum_{n=1}^{\infty} \frac{1}{(n+1)(n+2)} \):

• The sequence of partial sums, \( S_N = \frac{1}{2} - \frac{1}{N+2} \), helps us understand the limit behavior.
• As \( N \to \infty \), \( \frac{1}{N+2} \to 0 \), leaving \( \frac{1}{2} \).

Since this infinite series converges, we conclude that its partial sums approach the finite number \( \frac{1}{2} \). This indicates the sum of the series stabilizes at this value, rather than moving indefinitely or not settling.