91Ó°ÊÓ

When dealing with series, an essential aspect to understand is whether they converge or diverge. A series converges if, as you sum up its infinite terms, you approach a finite value. In simple terms, it means that the total amount doesn't explode to infinity or go haywire, but instead settles down to something sensible and simple.
When a series diverges, it does precisely the opposite. Instead of approaching a single, finite value, it grows without bound or doesn't settle into any particular pattern. This behavior makes the series impossible to pin down to a single number.
It's crucial to understand this distinction when evaluating series like the one in our problem. We needed to check whether the sum of the infinite terms results in a number we can call a sum or if it leads to chaos and divergence.

The concept of a partial sum is vital in tackling series problems. So, what exactly is a partial sum? Well, it's just the total you get when adding up a certain number of terms from the start of your series.
Think of it as a snapshot of the summation process. Instead of looking at the entire series, we only consider the first few terms to build up the sequence. In mathematics, we often denote this as \( S_n \), which means the sum up to the \( n \)th term of the series.
In our exercise, the series was given by \( \sum_{n=1}^{\infty} (\tan(n) - \tan(n-1)) \). Thanks to its telescoping nature, most of the middle terms canceled out, leaving us with the neat formula \( S_n = \tan(n) - \tan(0) \). This form made it easier to evaluate what happens as \( n \) grows. So, by understanding partial sums, we can slowly extend our grasp from finite additions toward the infinite sum.

To determine whether or not our series converges, we evaluated the limit of the partial sum. Evaluating the limit means looking at what happens as we add more and more terms - essentially letting \( n \) grow infinitely large.
The limit we examined was \( \lim_{n \to \infty} (\tan(n) - \tan(0)) \). Since \( \tan(0) \) is zero, the limit simplifies to \( \lim_{n \to \infty} \tan(n) \). The crucial challenge here is understanding how \( \tan(n) \) behaves as \( n \) continues to infinity. Notice that \( \tan(n) \) does not settle down to any single number as \( n \) becomes large.
Instead, it undergoes periodic oscillation due to the periodic nature of the tangent function, which causes it to not converge. Therefore, the absence of a finite limit for the partial sums leads us to conclude that this specific series diverges. This step illustrates why limit evaluation is so important for understanding the long-term behavior of a series.