Question

For the following telescoping series, find a formula for the nth term of the sequence of partial sums \(\left\\{S_{n}\right\\} .\) Then evaluate lim \(S_{n}\) to obtain the value of the series or state that the series diverges. \(^{n \rightarrow \infty}\) $$\sum_{k=1}^{\infty}\left(\frac{1}{k+2}-\frac{1}{k+3}\right)$$

Short answer

Answer: The value of the telescoping series converges to \(\frac{1}{3}\).

Step-by-step solution

Write out the telescoping series

Write the given telescoping series: $$\sum_{k=1}^{\infty}\left(\frac{1}{k+2}-\frac{1}{k+3}\right)$$

Compute the nth partial sum

To compute the nth partial sum \(S_{n}\), we need to find the sum of the first n terms of the series: $$S_{n} = \left(\frac{1}{1+2} - \frac{1}{1+3}\right) + \left(\frac{1}{2+2} - \frac{1}{2+3}\right) + \cdots + \left(\frac{1}{n+2} - \frac{1}{n+3}\right)$$ Notice that in each term after the first, the second term in the parentheses cancels out with the first term in the next parentheses. Applying the cancellation, we find: $$S_{n} = \frac{1}{3} - \frac{1}{n+3}$$

Find the limit of the partial sum as n approaches infinity

Now, we need to find the limit of the partial sum as n approaches infinity: $$\lim_{n \rightarrow \infty} S_{n} = \lim_{n \rightarrow \infty} \left(\frac{1}{3} - \frac{1}{n+3}\right)$$ As n approaches infinity, the term \(\frac{1}{n+3}\) approaches 0. Therefore, the limit becomes: $$\lim_{n \rightarrow \infty} S_{n} = \frac{1}{3} - 0 = \frac{1}{3}$$ Thus, the value of the series converges to \(\frac{1}{3}\).

Key concepts

Partial Sum

A partial sum is essentially an interim step in understanding a series. In mathematical terms, it refers to the sum of a certain number of initial terms in a series. Imagine you have a long list of numbers—perhaps infinite—and you want to find the sum of just the first few. This sum up to a given number of terms, say the first n terms, is called the partial sum, denoted as \( S_n \).
In the context of a telescoping series like \( \sum_{k=1}^{\infty}(\frac{1}{k+2} - \frac{1}{k+3}) \), calculating a partial sum simplifies significantly because many of the terms cancel out with each other. This is a characteristic property of telescoping series. For example, when you start adding the terms, you'll notice that each fractional part subtracts out with a part from a subsequent fraction, leaving behind only a few pieces.
In this particular series, each pair of terms like \( \frac{1}{k+2} - \frac{1}{k+3} \) almost entirely cancels out the previous and next pair, resulting in the partial sum formula \( S_n = \frac{1}{3} - \frac{1}{n+3} \). This shows how much remains of the series sum after adding up to the nth term.

Series Convergence

The concept of convergence in a series is about whether the sum of an infinite series eventually approaches some specific, finite value. If a series converges, it means that as you add more and more terms, the total sum gets closer and closer to a particular number, even though you're adding up infinitely many terms.
For the series \( \sum_{k=1}^{\infty}(\frac{1}{k+2} - \frac{1}{k+3}) \), we determine convergence by evaluating the limit of the partial sum \( S_n \) as \( n \) approaches infinity. In our example, as \( n \to \infty \), the term \( \frac{1}{n+3} \) becomes negligible (approaches zero), allowing the partial sum \( S_n \) to converge to a specific value, which is \( \frac{1}{3} \). Hence, the series is said to converge to \( \frac{1}{3} \).
This is an important characteristic of series, as it helps us identify if infinite series are meaningful in terms of giving a concrete total sum, despite their infinite length.

Limit of a Sequence

The limit of a sequence is a fundamental idea that helps us understand the behavior of sequences as they progress towards infinity. A sequence is essentially a list of numbers arranged in a specific order, and each number corresponds to a certain term \( n \).
The limit of a sequence \( \lim_{n \to \infty} a_n \) is the value that the terms of the sequence approach as \( n \) gets larger and larger. In the case of a partial sum sequence, finding this limit can tell you the value that the entire series converges to, if it converges at all.
In our telescoping series, the expressional form \( \frac{1}{3} - \frac{1}{n+3} \) reveals that as \( n \to \infty \), the term \( \frac{1}{n+3} \) trends towards zero. Thus, the entire sequence of partial sums approaches \( \frac{1}{3} \).
This conclusion underscores the elegance of limits: even with infinitely many terms, we can pinpoint a precise number they tend to settle to, highlighting both the power and precision of calculus in analyzing infinite processes.