91Ó°ÊÓ

A partial sum in a series is the sum of the first few terms, up to the th term. It provides a way to estimate the total sum of an infinite series by looking at a finite part of it. Understanding partial sums helps in analyzing whether an infinite series converges or diverges.
For a telescoping series like in our exercise, the partial sum has a specific pattern. Each term cancels out a neighboring term, simplifying the calculation. In our example, the partial sum for the first terms is given by:

• Calculate the sum up to the th term: \[ S_n = a_1 - a_{n+1} \] where \( a_n = \cos^{-1}\left( \frac{1}{n+1} \right) \).

This formula makes it easier to calculate the partial sum quickly. As you observe more terms cancel, leaving only the first and the last terms.

In the context of infinite series, understanding convergence and divergence is essential. A series converges if the partial sums approach a finite limit as you add more terms.
Conversely, a series diverges if there is no finite limit. Some series might increase indefinitely or oscillate without settling.

• Convergence: When the limit of partial sums is finite, the series converges. It means the sum tends to a specific value. In our example, the limit \( \lim_{n \to \infty} S_n = -\frac{\pi}{6} \) shows convergence.
• Divergence: If no finite limit exists, the series diverges. The sums could go to infinity or have no pattern to settle.

By analyzing the behavior of the partial sums, we determined our series converges with a sum of \(-\frac{\pi}{6}\). This tells us the series has a meaningful sum despite running to infinity.

The arccosine function, denoted as \( \cos^{-1}(x) \), is the inverse of the cosine function. It returns the angle whose cosine value is \( x \). Understanding this function is crucial in our exercise since it dictates the series terms. The range of arccosine is from \( 0 \) to \( \pi \), ensuring the output is an angle.
Let’s look at how it worked in the problem:

• The arccosine values, such as \( \cos^{-1}\left( \frac{1}{n+1} \right) \), form the building blocks of the telescoping series.
• We find specific values like \( a_1 = \cos^{-1}\left( \frac{1}{2} \right) = \frac{\pi}{3} \), providing a starting point.

For practical calculations, remember the property \( \cos(\theta) = x \rightarrow \theta = \cos^{-1}(x) \). In our series, this served to calculate and simplify terms effectively.
As you use the arccosine function, think about its inverse nature and range, giving you a geometrical perspective on angle measurements.