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. \(^{n \rightarrow \infty}\) $$\sum_{k=0}^{\infty}\left[\sin \left(\frac{(k+1) \pi}{2 k+1}\right)-\sin \left(\frac{k \pi}{2 k-1}\right)\right]$$

Short answer

If it converges, what is the value of the series? Answer: The given telescoping series converges, and its value is 0.

Step-by-step solution

Rewrite the Telescoping Series as Partial Sums

The telescoping series can be expressed as a sequence of partial sums by summing the terms of the series from k=0 to k=n: $$S_n = \sum_{k=0}^{n}\left[\sin \left(\frac{(k+1) \pi}{2 k+1}\right)-\sin \left(\frac{k \pi}{2 k-1}\right)\right]$$

Calculate First Few Partial Sums

In order to find a general formula for the nth term of the sequence of partial sums, we can calculate the first few partial sums to identify a pattern: For n = 0: $$S_0 = \sin\left(\frac{\pi}{1}\right) - \sin\left(\frac{0}{-1}\right) = 0$$ For n = 1: $$S_1 = \left[\sin\left(\frac{3\pi}{3}\right) - \sin\left(\frac{\pi}{1}\right)\right] + \left[\sin\left(\frac{\pi}{1}\right) - \sin\left(\frac{0}{-1}\right)\right] = 0$$ For n = 2: $$S_2 = \left[\sin\left(\frac{5\pi}{5}\right) - \sin\left(\frac{3\pi}{3}\right)\right] + \left[\sin\left(\frac{3\pi}{3}\right) - \sin\left(\frac{\pi}{1}\right)\right] + \left[\sin\left(\frac{\pi}{1}\right) - \sin\left(\frac{0}{-1}\right)\right] = 0$$ We can observe that for every term, the sine values in each term cancel out with the sine values in the previous term. This pattern continues, leading to: $$S_n = 0, \forall n \in \mathbb{N}$$

Evaluate Limit as n Approaches Infinity

Finally, we need to evaluate the limit of the sequence of partial sums as n approaches infinity: $$\lim_{n\to\infty} S_n = \lim_{n\to\infty} 0 = 0$$ Since the limit exists and is equal to 0, the series converges, and its value is 0: $$\sum_{k=0}^{\infty}\left[\sin \left(\frac{(k+1) \pi}{2 k+1}\right)-\sin \left(\frac{k \pi}{2 k-1}\right)\right] = 0$$

Key concepts

Partial Sums

In mathematics, partial sums are used to describe a portion of a larger series. By calculating the sum of the first several terms of a series, we can gain insights into its properties. The term "partial sums" refers to the sums obtained when you only add consecutive terms up to a certain point in the series. In this particular problem, we use partial sums to analyze a telescoping series—a type of series where many terms cancel out.

A partial sum helps us to simplify complex series by breaking down the addition process into separate, manageable pieces. By observing how these sums behave as we include more terms, we can explore the overall behavior of the series.

**Example:**
The sequence of partial sums for our series after including a few terms reveals a pattern:

• For \( n = 0 \), \( S_0 = 0 \)
• For \( n = 1 \), \( S_1 = 0 \)
• For \( n = 2 \), \( S_2 = 0 \)

This pattern suggests that the values might reach a stable point.

Sequence of Partial Sums

A sequence of partial sums is a sequence created by taking each partial sum from a series as a term in a new sequence. When looking at our series, each partial sum \( S_n \) is one of these terms. The entire sequence of partial sums can give clues about the convergence or divergence of a series.

In simpler terms, a sequence of partial sums lets us track the cumulative total of added terms as we progress through a series. By forming this sequence, we can study the behavior of the series in more detail and predict its long-term behavior.

When we analyze our series, the sequence \( \{S_n\} \) quickly becomes apparent:

• Each partial sum gradually stabilizes at 0.
• This repetition and simplicity indicate a possible convergence to 0.

Creating a sequence of partial sums is like constructing a roadmap, showing us each step of the journey through the series.

Convergence

Convergence in the context of a series is when the series approaches a specific value as more terms are added. For a sequence of partial sums, we say this sequence converges if it approaches a finite number as we consider more terms.

To determine convergence, we examine the behavior of the series as \( n \) approaches infinity. If the sequence of partial sums settles down to a particular value, the series converges.

For our telescoping series:

• The partial sums \( \{S_n\} \) were consistent at 0.
• As \( n \to \infty \), we evaluate \( \lim_{n\to\infty} S_n = 0 \).

The unchanging sum suggests convergence, confirming that our series does indeed reach a total sum of 0.

Series Evaluation

Series evaluation is the process of finding the total value that a given series converges to, if it converges at all. This involves calculating the limit of the sequence of partial sums.

In our telescoping series example, the series evaluation reveals that the series converges to 0. This is determined by:

• Writing out the partial sums to observe their pattern.
• Identifying cancellation among terms in each partial sum.
• Calculating the limit \( \lim_{n\to\infty} S_n = 0 \).

Every term beyond a certain point cancels with another, leaving a net total of zero. Therefore, the complete evaluation tells us that the infinite sum of this series is 0.

Series evaluation is essential in many areas of mathematics as it allows us to work with infinite numbers in a meaningful way.