91Ó°ÊÓ

A partial sum is essentially the summation of the initial segment of a series. In the context of an **infinite series**, we often start by considering a finite number of terms, say the first **N** terms, to grasp the behavior of the series. This finite sum is known as the partial sum, denoted as \( S_N \).
In our problem, the partial sum is given by the formula \( S_N = \frac{N+1}{N+4} \). This expression helps us understand how the sum of the series grows as we incorporate more terms.
It provides insight into the overall structure of the series and allows us to later calculate the total sum for an infinite number of terms, should that be possible.

The concept of a limit is crucial in determining the sum of an infinite series. When analyzing a partial sum \( S_N \), we aim to find its limit as \( N \) approaches infinity.
This means we're interested in the value that \( S_N \) approaches when we include infinitely many terms in the series. By understanding this limit, we can conclude the behavior or sum of the entire series.
In the given problem, simplifying the expression \( \frac{N+1}{N+4} \) and evaluating \( \lim_{N \to \infty} \frac{N+1}{N+4} \), we arrive at 1. Thus, the limit of the partial sum is 1, indicating that the sum of the infinite series converges to this value.

To say that a series converges means that as you add more and more terms, the sum approaches a specific finite value. This convergence is determined by the limit of its partial sums.
If the limit exists and is finite, the series is convergent. Otherwise, it's either divergent or conditioned on additional criteria.
In our solution, the infinite series sum \( \sum_{n=1}^{\infty} a_n \) converges because the partial sums, \( S_N = \frac{N+1}{N+4} \), have a limit as \( N \to \infty \) that is finite. The limit is calculated to be 1. Therefore, the series converges to this number, meaning the total sum of the series doesn't grow endlessly but stabilizes at this value of 1.

Calculus provides the foundational tools necessary to understand and work with infinite series. The concept of a limit, a fundamental idea in calculus, is what allows us to deal with series that involve infinitely many terms.
Using calculus, we analyze how functions behave as they approach certain points, which is instrumental in identifying the limits of sequences and partial sums.
In the exercise, we utilized calculus by applying the limit concept to the partial sum formula \( S_N = \frac{N+1}{N+4} \). By evaluating the limit of this expression as \( N \to \infty \), we conclude that the series converges to 1. Thus, calculus allows us to rigorously demonstrate that the infinite series settles at a summed value, bringing clarity to the behavior of the series over an infinite span.