Problem 39 - Thomas Calculus

Question

In Exercises \(35-40,\) 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. $$ \sum_{n=1}^{\infty}\left(\cos ^{-1}\left(\frac{1}{n+1}\right)-\cos ^{-1}\left(\frac{1}{n+2}\right)\right) $$

Step-by-Step Solution

Answer

The series diverges; the partial sum does not converge to a definite value.

Step 1: Understanding Partial Sums

The series is given by \( \sum_{n=1}^{\infty}\left(\cos ^{-1}\left(\frac{1}{n+1}\right)-\cos^{-1}\left(\frac{1}{n+2}\right)\right) \), which suggests a telescoping series, where many terms cancel each other out when summed over a sequence.

Step 2: Write the General Term

The general term of the series can be written as \( b_n = \cos^{-1}\left(\frac{1}{n+1}\right) - \cos^{-1}\left(\frac{1}{n+2}\right) \). This represents a difference of inverse cosine functions which will result in cancellation in a partial sum.

Step 3: Establish the Partial Sum Formula

The \( n \)th partial sum \( S_n \) is given by adding up the first \( n \) terms of the sequence: \[ S_n = \left(\cos^{-1}(1) - \cos^{-1}\left(\frac{1}{n+2}\right)\right). \]Most of the intermediate terms \( \cos^{-1}\left(\frac{1}{k+1}\right) \) and \( -\cos^{-1}\left(\frac{1}{k+2}\right) \) cancel out when computed in sequence.

Step 4: Analyzing Limit of Partial Sums

Evaluate \( \lim_{n \to \infty} S_n \). As \( n \) grows, \( \cos^{-1}\left(\frac{1}{n+2}\right) \) approaches \( \cos^{-1}(0) = \frac{\pi}{2} \) since \( \frac{1}{n+2} \) approaches zero.

Step 5: Conclusion on Convergence

The partial sum approaches:\[ \lim_{n \to \infty} S_n = \cos^{-1}(1) - \frac{\pi}{2} = 0 - \frac{\pi}{2} = -\frac{\pi}{2}, \]indicating the infinite series diverges as it does not converge to a finite number.

Key Concepts

Series Convergence

Series convergence is a concept that helps us determine whether the sum of infinite terms leads to a finite value or not. For a series to converge, its sequence of partial sums must approach a specific limit as more terms are added.
In simpler terms, if you keep adding terms from an infinite series, the total should settle down to a number. Otherwise, it diverges.

• In convergent series: Partial sums approach a fixed numerical limit.
• In divergent series: Partial sums do not settle and keep increasing or keep oscillating.

Determining convergence involves checking the limit of the partial sum as queried in our exercise. In the given series, the indicators leaned towards divergence as the sequence of partial sums did not approach a finite value, finally resulting in \(-\frac{\pi}{2}\), indicating it does not settle on a precise number.

Inverse Trigonometric Functions

Inverse trigonometric functions help us find angles given their trigonometric ratios. For instance, inverse cosine, denoted as \( \cos^{-1}\), presents the angle whose cosine is a certain value. These functions are essential in various mathematical contexts, including series like the one in our exercise.
Here's a quick look at inverse cosine characteristics:

• The range of \( \cos^{-1} \): From \( 0 \) to \( \pi \).
• Determines the principal angle for which \( \cos(\theta) = x \).
• Facilitates the computation within mathematical sequences and series.

In the problem we're exploring, \( \cos^{-1}\) played a role in facilitating the formation of a telescoping series. Through its properties, it allowed terms to cancel each other across the series, revealing the underlying structure.

Partial Sums

A partial sum is composed of the initial portion of an infinite sequence or series. It aids in approximating and eventually understanding the overall behavior of the full series.
For a series \( \sum a_n\), the nth partial sum, \( S_n\), is the summation of the first \( n\) terms: \[S_n = a_1 + a_2 + a_3 + \dots + a_n\].

• Partial sums highlight which parts of a series eventually cancel each other, as seen in telescoping series.
• They act as a crucial stepping stone in examining convergence and divergence.

In our example, partial sums were instrumental in establishing the series as a telescoping one. Upon evaluating the nth partial sum: \( S_n = \cos^{-1}(1) - \cos^{-1}\left(\frac{1}{n+2}\right) \), it was clear how intermediate terms in the summation canceled out, impacting overall understanding of the series' behavior.