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Chapter 12: Problem 5

Determine whether the series converges or diverges. $$\sum \frac{k !}{100^{k}}$$

Short Answer

Expert verified
The given series is \(\sum \frac{k !}{100^{k}}\). Applying the Ratio Test, we find the ratio of consecutive terms as \(\frac{k+1}{100}\). The limit of this ratio as \(k\) approaches infinity is infinite, which is greater than 1. Therefore, the series diverges according to the Ratio Test.

Step by step solution

01

Understanding the Ratio Test

The Ratio Test states that for a series \(\sum a_k\), if the limit \[\lim_{k \to \infty} \left|\frac{a_{k+1}}{a_k}\right|\] exists and is less than 1, the series converges. If the limit is greater than 1, the series diverges. If the limit is equal to 1, the test is inconclusive.
02

Apply the Ratio Test to the given series

Our given series is: \[\sum \frac{k !}{100^{k}}\] Now let's find the ratio of consecutive terms: \[\left|\frac{a_{k+1}}{a_k}\right| = \left|\frac{(k+1)!}{100^{k+1}} \cdot \frac{100^k}{k!}\right|\]
03

Simplify the expression

Simplifying the expression, we get: \[\left|\frac{(k+1)!}{100^{k+1}} \cdot \frac{100^k}{k!}\right| = \frac{(k+1)(k!)100^k}{100^{k+1}k!}\]
04

Cancel out common terms

Cancel out \(k!\) and \(100^k\) from the expression: \[\frac{(k+1)(k!)100^k}{100^{k+1}k!} = \frac{k+1}{100}\]
05

Calculate the limit

Now we need to find the limit of the ratio as k approaches infinity: \[\lim_{k \to \infty} \frac{k+1}{100}\]
06

Analyze the limit

Since the limit is infinite (greater than 1), the series diverges according to the Ratio Test. Therefore, the given series \(\sum \frac{k !}{100^{k}}\) diverges.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ratio Test
The Ratio Test is a powerful tool to determine the convergence or divergence of an infinite series. It's particularly useful for series involving factorials and exponential terms.
This test uses a simple limit calculation to make the determination.
For a given series, say \[\sum a_k,\]we compute the limit\[\lim_{k \to \infty} \left|\frac{a_{k+1}}{a_k}\right|.\]
  • If this limit is less than 1, the series converges.
  • If it is greater than 1, the series diverges.
  • If equal to 1, the test is inconclusive.
The Ratio Test can be thought of as comparing the growth rates of consecutive terms to the convergence threshold of 1.
Limit Calculation
Limit calculation is central to applying the Ratio Test. In this exercise, once the ratio of consecutive terms is found, simplifying and calculating its limit is key.
The expression \[\frac{k+1}{100}\]is a straightforward example. As \(k\) approaches infinity, calculating the limit:\[\lim_{k \to \infty} \frac{k+1}{100}\]means evaluating whether the fraction changes with growing \(k\).
Since \(k\) can grow indefinitely, dividing an infinitely large number by 100 still results in infinity, certainly exceeding the threshold value of 1.
The result informs us about the behavior of the series, confirming divergence due to exceeding 1.
Factorials in Series
Factorials, denoted as \(k!\), are crucial in many infinite series and pose unique challenges in limits and convergence assessments.
Each factorial \(k!\) grows quite rapidly, as it is the product of all integers from 1 to \(k\).
This rapid growth often leads to divergence unless tempered by very large denominators.
In this case, the factorial's explosion in size relative to a power base like 100 means that the numerator grows much faster than the exponential denominator, a common characteristic of divergent behavior when unchecked. This imbalance is what drives the divergence in the example given.
Infinite Series
An infinite series is a sum of an infinite sequence of terms, commonly denoted as \[\sum a_k.\] These series can either converge to a finite number or diverge, growing indefinitely.
An infinite series's behavior is determined by its terms鈥� properties, such as growth rate driven by factors like exponential components or factorial parts.
Tools like the Ratio Test help understand these properties. The series \[\sum \frac{k!}{100^k}\]initially seems daunting due to its factorial and power components, but through the Ratio Test, it's shown to diverge.
Understanding series properties allows for analytical approaches to challenges in mathematical sequences and infinite sums.

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