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The Limit Comparison Test is a nifty tool for determining the convergence or divergence of an infinite series.
It involves comparing a series you are interested in, say \( \sum a_n \), with another known series, \( \sum b_n \). Here's how it works:

• Calculate \( \lim_{n \to \infty} \frac{a_n}{b_n} \).
• If this limit is a positive finite number, then both series \( \sum a_n \) and \( \sum b_n \) either converge together, or diverge together.

This test is particularly useful when the terms of your series complicated or when they approximate the terms of a simpler known series. In the given exercise, it immediately shows us whether the series behaves like the harmonic series, which is a known diverging series.

The Harmonic Series is a classic example in the study of convergence. It is given by:\[ \sum_{n=1}^{\infty} \frac{1}{n} \].Though the terms \( \frac{1}{n} \) get smaller as \( n \) grows, they do not decrease fast enough for the series to converge.
In fact, the Harmonic Series diverges.
The divergence of this series can often be proven using the integral test or the limit comparison test. Because of its straightforward terms, the harmonic series is frequently used as a benchmark for other series in comparison tests. If the terms of a complex series can be shown to grow like the harmonic series, then we can easily determine the convergence behavior by observing the harmonic series.

The Tangent Function is a familiar concept from trigonometry, often written as \( \tan(x) \). It relates to sine and cosine:\[ \tan(x) = \frac{\sin(x)}{\cos(x)} \]. For very small values of \( x \), the function behaves approximately like \( x \) itself, which is crucial for series involving tangent.
As in the exercise, for small inputs such as \( \frac{1}{n} \), we use the approximation:\[ \tan\left(\frac{1}{n}\right) \approx \frac{1}{n} \],which simplifies the problem significantly by allowing us to compare it to the harmonic series. This approximation is the key reason why the Limit Comparison Test can be applied, indicating that the given tangent series diverges.

An Infinite Series is essentially the sum of infinitely many numbers listed in a sequence.
Formally, for a series \( \sum_{n=1}^{\infty} a_n \), this means trying to find out whether adding an infinite number of terms gives you a finite result or not.

• If as you add more and more terms the sum gets closer and closer to a specific number, the series is said to converge.
• If the sum keeps growing or fluctuates without settling on a number, the series diverges.

Understanding whether a series converges or diverges is integral to many areas of mathematics and its applications. Techniques like the Limit Comparison Test help us make these determinations, as with the series involving \( \tan \left( \frac{1}{n} \right) \), which was shown to diverge like the harmonic series it was compared against.