Question

Evaluate the following telescoping series or state whether the series diverges. $$ \sum_{n=1}^{\infty}(\sin n-\sin (n+1)) $$

Short answer

The series diverges.

Step-by-step solution

Identify the Series

We begin by examining the series given: \( \sum_{n=1}^{\infty} (\sin n - \sin (n+1)) \). This is a telescoping series, where consecutive terms cancel each other out.

Write Out Initial Terms

To see the telescoping nature, write out the initial terms: \( (\sin 1 - \sin 2) + (\sin 2 - \sin 3) + (\sin 3 - \sin 4) + \ldots \).We see that most terms cancel: +\(\sin 2\) from one term cancels with -\(\sin 2\) from the next term, etc.

Evaluate the Pattern

Notice that this pattern continues indefinitely, leaving only the very first term and the negative of the term after the last visible one.For a general finite version of the series from 1 to \( k \), it would look like:\( \sin 1 - \sin (k+1) \).

Analyze the Limit

As \( k \) approaches infinity, the expression given by the sum \( \sin 1 - \sin (k+1) \) is considered.Since \( -1 \leq \sin(x) \leq 1 \) for all \( x \), as \( k \to \infty \), \( \sin(k+1) \) does not converge to a specific number but oscillates.Thus, the overall limit is undefined, and the series does not converge.

Conclusion about Convergence

The series \( \sum_{n=1}^{\infty} (\sin n - \sin (n+1)) \) diverges because the limit of its partial sums is non-existent as \( k \to \infty \).

Key concepts

Series Evaluation

When faced with evaluating a series, particularly a telescoping one, it's crucial to identify the structure of the series first. A telescoping series is a special type of series where consecutive terms cancel each other. This leads to simpler forms that are easy to evaluate. Consider the series \( \sum_{n=1}^{\infty} (\sin n - \sin (n+1)) \). Here, by writing out the initial terms, we can observe the cancellation pattern:

• \((\sin 1 - \sin 2) + (\sin 2 - \sin 3) + (\sin 3 - \sin 4) + \ldots\)

Each positive term cancels with the next negative term, simplifying the series significantly. When evaluating such series, you're often left with the first term and the last term from a finite segment. For a segment from 1 to \( k \), the series simplifies to \( \sin 1 - \sin (k+1) \). This simplification is key in determining the behavior of the series as \( k \) approaches infinity.

Divergence

In the context of series, divergence refers to the failure of a series to converge to a specific limit. For a telescoping series like \( \sum_{n=1}^{\infty} (\sin n - \sin (n+1)) \), though many terms cancel out, the issue of divergence arises from the behavior of the remaining terms as \( k \to \infty \).

Series Limit Analysis
The expression simplifies to \( \sin 1 - \sin (k+1) \) as the number of terms grows. Here, the term \( \sin (k+1) \) oscillates between -1 and 1 for all values of \( k \), never settling to a specific limit. This oscillation is the core reason why the series lacks a well-defined convergence.

• A series converges if it approaches a specific finite limit as more terms are added.
• If it oscillates or tends towards infinity without settling, it diverges.

Therefore, due to this ongoing oscillation and lack of settling to a single number, this telescoping series diverges.

Infinite Series

Understanding infinite series entails recognizing that they are sums of infinitely many terms. In our example, we deal with an infinite series: \( \sum_{n=1}^{\infty} (\sin n - \sin (n+1)) \). Here are some key points:

• Nature: "Infinite" implies that there is no endpoint; the process theoretically continues forever.
• Behavior: For convergence, an infinite series should approach a finite number as more terms are added. For telescoping series, much depends on the residue, or what's left after cancellation.

For the series in question, even though terms cancel widely, leading some to quickly assume it might converge, it's crucial to evaluate the remaining terms. The oscillation of \( \sin (k+1) \) over an infinite range with no fixed point invalidates convergence.

Evaluating infinite series correctly requires careful analysis of term behavior as they tend towards infinity, considering whether it stabilizes or not.