91Ó°ÊÓ

When we talk about partial sums, we are referring to the sum of the first few terms of an infinite series. For a given series \ \( \left\{ a_k \right\} \) , the partial sum, \ \( S_n \) , includes terms from \ \( a_1 \) to \ \( a_n \) . It serves as a way to approximate the sum of the entire series by considering only a finite portion of it.

In our exercise, the sequence of partial sums is denoted by \ \( S_n = \sum_{k=1}^{n} \left( \frac{1}{k+2} - \frac{1}{k+3} \right) \) . Initially, this may seem complex, but simplifying often involves noticing patterns or cancellations within the series.

A useful strategy for simplifying the partial sum is recognizing **telescoping**. In telescoping series, many terms cancel out, significantly simplifying the sum. This aspect focuses on reducing a potentially complicated sum to just a few terms, as seen when \ \( S_n = \frac{1}{3} - \frac{1}{n+3} \) after cancellation.

The concept of series convergence revolves around whether the sum of an infinite series approaches a specific finite number. To determine this, we look at the limit of the sequence of partial sums as \ \( n \) approaches infinity. If this limit exists and is finite, the series is said to converge. Otherwise, it diverges.

In our problem, after simplifying the partial sum \ \( S_n = \frac{1}{3} - \frac{1}{n+3} \) , we notice that as \ \( n \to \infty \) , the term \ \( \frac{1}{n+3} \) approaches zero. This results in the limit \ \( \lim_{n \to \infty} S_n = \frac{1}{3} \) .

This convergence suggests that the infinite series approaches the sum of \ \( \frac{1}{3} \) . Therefore, evaluating the convergence of series revolves around knowing when and how the infinite addition of terms stabilizes to a particular value.

The limit of a sequence is a fundamental concept in calculus. It involves finding what value a sequence approaches as the index number (often denoted as \ \( n \) ) goes to infinity. Understanding this helps in determining the behavior of sequences and series over the long term.

For the exercise at hand, examining the sequence of partial sums \ \( S_n = \frac{1}{3} - \frac{1}{n+3} \) helps us understand the eventual behavior. As \ \( n \to \infty \) , the fraction \ \( \frac{1}{n+3} \) becomes increasingly smaller, effectively reaching zero.

This is due to the denominator \ \( n+3 \) growing without bound, pulling \ \( \frac{1}{n+3} \) closer and closer to zero. Therefore, \ \( S_n \) consistently approaches the constant \ \( \frac{1}{3} \) , showcasing how the limit affirms series convergence and defines the sum for an infinite series.