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Chapter 10: Problem 38

Find a formula for the \(n\) th partial sum of the series and use it to determine if the series converges or diverges. If a series converges, find its sum. $$\sum_{n=1}^{\infty}(\tan (n)-\tan (n-1))$$

Short Answer

Expert verified
The series diverges because the partial sums do not approach a finite limit.

Step by step solution

01

Understanding the Series

The given series is \( \sum_{n=1}^{\infty} (\tan(n) - \tan(n-1)) \). We need to find the \( n \)th partial sum of this series and determine if it converges or diverges.
02

Identify the Telescoping Nature of the Series

Observe that in the expression \( (\tan(n) - \tan(n-1)) \), each term \( \tan(n) \) is subtracted by the previous term \( \tan(n-1) \). This indicates that the series is telescoping, meaning that most terms will cancel out in the partial sum.
03

Writing the Partial Sum

The partial sum \( S_n \) for the first \( n \) terms of the series is given by:\[S_n = (\tan(1) - \tan(0)) + (\tan(2) - \tan(1)) + \cdots + (\tan(n) - \tan(n-1))\] Notice that this can be rewritten as\[S_n = \tan(n) - \tan(0)\]Thus, the \( n \)th partial sum is \( \tan(n) - \tan(0) \).
04

Examine Convergence of the Series

To determine if the series converges, examine the limit of \( S_n \) as \( n \rightarrow \infty \). We have:\[\lim_{n \to \infty} S_n = \lim_{n \to \infty} (\tan(n) - \tan(0))\] The function \( \tan(n) \) does not have a finite limit as \( n \to \infty \) because the tangent function periodically reaches \( \infty \) or \( -\infty \). Therefore, \( S_n \) does not converge to a finite value.
05

Conclusion about the Series

Since the limit of the partial sum \( S_n \) as \( n \to \infty \) does not exist, the series \( \sum_{n=1}^{\infty}(\tan(n)-\tan(n-1)) \) diverges.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Partial Sums
A partial sum is a significant concept when dealing with series. It refers to the sum of the first "n" terms of a sequence. In the context of our given series, the partial sum helps us see how the series behaves as more terms are added. For the series \( \sum_{n=1}^{\infty} (\tan(n) - \tan(n-1)) \), the partial sum \( S_n \) is expressed as the total of the first \( n \) terms:
  • \( S_n = (\tan(1) - \tan(0)) + (\tan(2) - \tan(1)) + \cdots + (\tan(n) - \tan(n-1)) \)
  • This simplifies to \( S_n = \tan(n) - \tan(0) \) because the intermediate terms cancel each other out.
The concept of telescoping plays a crucial role here, allowing us to simplify the expression by cancelling terms of similar magnitude. This makes it efficiently manageable to find the form of the nth partial sum.
Series Convergence
Series convergence is a fundamental idea when analyzing whether a series approaches a finite limit as more terms are added. For a series to converge, its sequence of partial sums must approach a specific value. Consider the partial sum for the series \( S_n = \tan(n) - \tan(0) \). We explore:
  • \( \lim_{n \to \infty} S_n = \lim_{n \to \infty} (\tan(n) - \tan(0)) \)
  • We observe that the function \( \tan(n) \) does not have a definitive limit as \( n \to \infty \).
This is because the tangent function oscillates between \( \infty \) and \( -\infty \) as it approaches its asymptotes. Therefore, the series in question does not approach a finite limit, leading us to question its convergence.
Divergent Series
Not every series converges; some are divergent. A divergent series is one where the sequence of partial sums does not tend to a particular limit. In the context of \( \sum_{n=1}^{\infty}(\tan(n)-\tan(n-1)) \), considering its telescoping nature, we already know:
  • The partial sum \( S_n = \tan(n) - \tan(0) \) illustrates that as \( n \rightarrow \infty \), \( \tan(n) \) doesn't settle at a consistent value.
  • Since \( \tan(n) \) oscillates indefinitely without reaching a stable state, \( S_n \) does not converge to a specific, finite number.
Thus, the divergence is inherent due to the unbounded behavior of the tangent function within this series. Recognizing a divergent series is crucial as it dictates that the series does not sum to a particular value, rendering it unable to approximate or converge.

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