91Ó°ÊÓ

The exponential series is a cornerstone of many mathematical concepts, serving as a fundamental expansion for exponential functions. It is represented by the formula:

• \[ e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} \]

This means that the exponential function can be expressed as an infinite sum of powers of the variable divided by factorials. This method of representation is advantageous, especially for calculations involving approximations of the exponential function.
In the context of the original exercise, one portion of the series is \(\sum_{n=1}^{\infty} \frac{4^n}{n!}\). This is a fragment of the exponential series, specifically for \(x = 4\). Therefore, by recognizing it as part of the series expansion of \(e^x\), we know that it converges. This convergence implies that the series approaches a specific value as \(n\) increases endlessly.

Factorials are a mathematical concept that display rapid growth with increasing \(n\). A factorial \(n!\) is the product of all positive integers up to \(n\). For instance:

• \(1! = 1\)
• \(2! = 2\)
• \(3! = 6\)

This swift escalation means that factorial denominators contribute to the convergence of series because they make the terms vanish quickly as \(n\) grows. In the given exercise, both components of the series have \(n!\) in the denominator, strongly influencing the speed at which terms shrink. This is crucial in testing for convergence since smaller terms aid the series in converging.

The ratio test is a popular method used to determine whether an infinite series converges or diverges. For a series \(\sum a_n\), the test involves always computing the limit:

• \[ \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \]

The criteria for the test are:

• If the limit is less than 1, the series converges.
• If the limit is greater than 1, the series diverges.
• If the limit is exactly 1, the test is inconclusive.

In the given task, we applied the ratio test to \(\sum_{n=1}^{\infty} \frac{n}{n!}\). The limit evaluated to 1, which makes the test inconclusive. However, even though inconclusive, this test confirms that we need to explore other methods, such as the comparison test, to conclude convergence.

The comparison test is a method to decide if a series converges by comparing it with another series whose convergence properties are known. In this test, we analyze a series \(\sum a_n\) by finding another series \(\sum b_n\) such that:

• If \(0 \leq a_n \leq b_n\) for all \(n\), and \(\sum b_n\) is convergent, then \(\sum a_n\) is also convergent.
• Conversely, if \(a_n \geq b_n\) and \(\sum b_n\) diverges, then \(\sum a_n\) diverges.

For the problem at hand, the series \(\sum_{n=1}^{\infty} \frac{n}{n!}\) was compared to \(\sum_{n=1}^{\infty} \frac{1}{n^2}\), which is a well-known convergent series. Since \(\frac{n}{n!}\) falls below these terms for larger \(n\), it allows us to reason that the series is convergent by comparison to the convergent series \(\frac{1}{n^2}\). This comparison provided the needed evidence to establish convergence when the ratio test was inconclusive.