Question

Is the series convergent or divergent? $$ \sum_{n=1}^{\infty} \frac{2^{n}}{n^{4}} $$

Short answer

The series is divergent because the ratio test gives a limit of 2 (>1).

Step-by-step solution

Identify the Series and Compare with Known Series

The given series is \( \sum_{n=1}^{\infty} \frac{2^{n}}{n^{4}} \). To determine convergence or divergence, it's helpful to compare it with a known series, like the p-series or geometric series. Here, the series doesn't obviously fall into either of these, so we consider the Ratio Test.

Apply the Ratio Test

The Ratio Test suggests that for \( \sum a_n \), if the limit \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L \) is less than 1, the series converges; if \( L > 1 \), it diverges; and if \( L = 1 \), the test is inconclusive. Set \( a_n = \frac{2^n}{n^4} \) and calculate:\[\lim_{n \to \infty} \left| \frac{2^{n+1}/(n+1)^4}{2^n/n^4} \right| = \lim_{n \to \infty} \left(2 \cdot \frac{n^4}{(n+1)^4}\right)\]

Simplify the Limit Expression

Simplify the expression from the Ratio Test:\[\lim_{n \to \infty} 2 \cdot \left(\frac{n^4}{(n+1)^4}\right)\]This simplifies to:\[2 \cdot \lim_{n \to \infty} \left(\frac{n}{n+1}\right)^4 = 2 \cdot \left(\frac{1}{1}\right)^4 = 2\]

Conclusion from the Ratio Test

Since the limit is \( L = 2 \), which is greater than 1, the series \( \sum_{n=1}^{\infty} \frac{2^n}{n^4} \) is divergent according to the Ratio Test.

Key concepts

Understanding the Ratio Test

The Ratio Test is an essential tool in determining the convergence or divergence of an infinite series. It's especially useful when dealing with series involving exponential terms or factorials. The basic idea is to compare each term in the series with the next term. Here's how it works:

• The test is applied to the absolute value of the ratio of consecutive terms: \( \frac{a_{n+1}}{a_n} \).
• If the limit \( L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \) exists, you make conclusions based on its value.
• If \( L < 1 \), the series converges absolutely.
• If \( L > 1 \), as in our problem, the series diverges.
• If \( L = 1 \), the test is inconclusive, and you might need to try other convergence tests.

This test is appreciated for its straightforwardness. However, it's paramount to remember it might not work on all series, particularly when \( L \) equals 1.

Delving into Infinite Series

An infinite series is a sum of infinite terms derived from a sequence of numbers. Mathematically, it's expressed as \( \sum_{n=1}^{\infty} a_n \). Understanding infinite series is crucial as they appear across various mathematical fields including calculus and real analysis.Infinite series can either converge or diverge:

• A convergent series approaches a specific value as more terms are added. This means the sum gets closer and closer to a finite limit.
• A divergent series does not approach a specific limit, meaning the sum grows indefinitely or keeps oscillating without settling.

There are numerous tests to determine the convergence or divergence of a series, including the Ratio Test, Root Test, and Integral Test among others. Each test might be suited to different types of series.

What is Divergence in Series?

Divergence refers to the behavior of an infinite series that doesn't settle on a fixed limit. When a series diverges, its terms do not sum to a finite value or repeat regularly around a value. Divergence can take several forms:

• When the terms of the series keep adding up to a very large number, indicating they're increasing without bound.
• In cases where the series' sum keeps oscillating between certain values without settling into convergence.
• Sometimes, the general term of the series doesn’t approach zero, which is an initial indicator of divergence.

Divergence isn't "bad" per se; it simply indicates that specific tools or methods need to be applied to understand the series' behavior. Understanding whether a series converges or diverges is crucial for applying mathematical principles correctly in real-world analyses.