Positive Series Convergence Determine Convergence Of Series N/(10n+12)

Hey there, math enthusiasts! Today, we're diving into the fascinating world of positive series convergence. We'll be tackling a classic problem: determining whether a given series converges or diverges. And to make things super clear, we'll be explicitly stating the tests we're using along the way. So, buckle up and let's get started!

(a) Diving Deep into the Series: $\sum_{n=1}^{\infty} \frac{n}{10 n+12}$

Alright, let's kick things off with the series $\sum_{n=1}^{\infty} \frac{n}{10 n+12}$. The big question is: does this series converge, or does it diverge? To figure this out, we need to pull out our trusty convergence tests. One of the first tests that often comes to mind when dealing with series that look like rational functions (polynomials divided by polynomials) is the Divergence Test, also known as the n-th Term Test.

The Divergence Test: A First Look

The Divergence Test is super straightforward. It states that if the limit of the terms of the series, $a_n$, as n approaches infinity is not equal to zero, then the series must diverge. In mathematical terms:

If $\lim_{n \to \infty} a_n \neq 0$, then $\sum_{n=1}^{\infty} a_n$ diverges.

It's crucial to understand that this test can only tell us if a series diverges. If the limit does equal zero, the test is inconclusive, and we need to try another method. Think of it like a first-line defense – if it fails, we move on to more sophisticated tools.

Applying the Divergence Test to Our Series

So, let's apply this to our series, $\sum_{n=1}^{\infty} \frac{n}{10 n+12}$. We need to find the limit of the terms:

$\lim_{n \to \infty} \frac{n}{10 n+12}$

To evaluate this limit, we can divide both the numerator and the denominator by the highest power of n, which in this case is n itself:

$\lim_{n \to \infty} \frac{n/n}{(10 n+12)/n} = \lim_{n \to \infty} \frac{1}{10 + 12/n}$

Now, as n approaches infinity, the term $12/n$ approaches zero. This leaves us with:

$\lim_{n \to \infty} \frac{1}{10 + 0} = \frac{1}{10}$

Aha! The limit is $\frac{1}{10}$, which is definitely not equal to zero. Therefore, according to the Divergence Test, the series $\sum_{n=1}^{\infty} \frac{n}{10 n+12}$ diverges.

Wrapping Up Part (a)

So, there you have it! By applying the Divergence Test, we've confidently determined that the series $\sum_{n=1}^{\infty} \frac{n}{10 n+12}$ diverges. Remember, the Divergence Test is a powerful initial tool in our convergence-testing arsenal. It's quick, easy, and often gives us a definitive answer right off the bat. But don't forget, if the limit is zero, we'll need to explore other tests to reach a conclusion.

Understanding the Big Picture of Divergence Test

Before moving on, let's reinforce why the Divergence Test works. Think about it intuitively: if the individual terms of a series don't approach zero, it means we're continually adding non-negligible amounts. If you keep adding amounts that don't shrink towards zero, the sum is bound to grow without bound, hence the series diverges. This is why the Divergence Test is such a fundamental concept in series analysis. It helps us quickly identify cases where the series is doomed to diverge because its terms don't diminish appropriately. Always remember to check this test first, as it can save you a lot of time and effort if it yields a non-zero limit. It’s like a preliminary check-up – a quick way to spot a major issue before diving into more complex diagnostics. In our case, $\sum_{n=1}^{\infty} \frac{n}{10 n+12}$, the terms approach $\frac{1}{10}$, a clear sign that we're adding values that don't get smaller, leading to divergence. So, mastering this test is crucial for efficiently analyzing series convergence.

Additional Notes on Divergence Test Applications

The beauty of the Divergence Test also lies in its simplicity and broad applicability. It doesn't require complex calculations or intricate comparisons. As long as you can compute the limit of the sequence, you can apply this test. This makes it incredibly useful as a first step in analyzing a wide range of series. For example, consider a series where the terms oscillate or approach a non-zero constant. The Divergence Test can immediately tell us that these series diverge. However, it's important to remember the test's limitation: it can only prove divergence. If the limit of the terms is zero, it doesn’t guarantee convergence. We'll need to turn to other tests like the Integral Test, Comparison Test, Ratio Test, or Root Test. Think of it as a screening process – it identifies the obvious cases of divergence but requires further investigation for cases that are more nuanced. This strategic approach to problem-solving is key to mastering series analysis. By starting with the Divergence Test, you can efficiently narrow down your options and focus on the appropriate tests for each scenario.

Practical Tips for Applying the Divergence Test

To effectively use the Divergence Test, there are a few practical tips to keep in mind. First, always write out the limit expression clearly. This helps you avoid mistakes and ensures you're focusing on the correct calculation. Second, familiarize yourself with common limit techniques, such as dividing by the highest power of n, L'Hôpital's Rule, and recognizing standard limit results. These skills will make evaluating limits much easier and faster. Third, be vigilant about the conditions for applying the test. The Divergence Test only applies to series where the terms are well-defined and the limit exists. Finally, practice makes perfect! The more you apply the Divergence Test to different series, the more comfortable and confident you'll become. You'll start to recognize patterns and quickly identify situations where the test is most effective. Remember, this test is a foundational tool, so mastering it will significantly enhance your ability to analyze series convergence. Keep practicing, and you'll find yourself efficiently determining divergence in many situations.

I hope this comprehensive explanation of the Divergence Test and its application to the series $\sum_{n=1}^{\infty} \frac{n}{10 n+12}$ has been helpful! Let's keep exploring the world of series convergence and uncover more fascinating techniques. Stay tuned for more insights and examples!