91Ó°ÊÓ

Let's explore how to determine the general term of a series from its partial sums. In many exercises involving series, we're given a sequence of partial sums, represented as \( S_n \), and our task is to derive the expression for the general term, \( a_n \). To do this, we use a simple formula:

• \( a_n = S_n - S_{n-1} \)

This formula involves taking the difference between consecutive partial sums. The idea is that the difference between successive terms in the sequence of partial sums gives us the general term of the series.
For example, with \( S_n = 2 - \frac{n+2}{2^n} \), subtract the previous partial sum from the current one:
\[ a_n = \left(2 - \frac{n+2}{2^n}\right) - \left(2 - \frac{n+1}{2^{n-1}}\right) \]
This allows us to isolate \( a_n \), eventually simplifying to \( a_n = \frac{n}{2^n} \). Understanding how to find \( a_n \) is crucial in predicting the behavior of the series.

Convergence in mathematics describes whether a series settles towards a specific value as its terms increase. Studying convergence helps us understand the behavior of a series in the long run. For a series with partial sums \( S_n \), we examine its limit as \( n \) approaches infinity.
In this case, if the terms of \( S_n \) shrink towards a finite value, we say that the series converges to that limit.

• The given series has partial sums: \( S_n = 2 - \frac{n+2}{2^n} \)

As \( n \to \infty \), notice that the exponential factor in the denominator, \( 2^n \), grows rapidly, causing the term \( \frac{n+2}{2^n} \) to tend towards zero.
As a result, the entire expression \( S_n = 2 - \frac{n+2}{2^n} \) trends towards \( 2 \).
This outcome tells us that the series converges, and its sum approaches \( 2 \), which is a crucial insight into understanding the long-term behavior.

The concept of limits is fundamental in calculus and analysis, applied to study sequences and series. A sequence approaches a particular value as its terms extend infinitely. We denote the limit of a sequence \( S_n \) as \( n \to \infty \), encapsulating this idea mathematically.

• For a partial sum \( S_n = 2 - \frac{n+2}{2^n} \), we're interested in its limit as \( n \to \infty \).

To determine this, we look at how the term \( \frac{n+2}{2^n} \) behaves as \( n \) increases. Since \( 2^n \) is exponential, it rises much quicker than the polynomial term \( n+2 \), forcing \( \frac{n+2}{2^n} \) to approach zero.
Hence, the limit of the sequence of partial sums \( S_n \) simplifies to \( \lim_{n \to \infty} S_n = 2 - 0 = 2 \). Understanding limits provides a deeper view of the long-term tendencies of sequences and enriches our comprehension of series convergence.