Question

Choose your test Use the test of your choice to determine whether the following series converge. $$\sum_{k=1}^{\infty} \frac{k^{100}}{(k+1) !}$$

Short answer

Answer: The ratio test was inconclusive with a limit of 1. To determine if the series converges or not, further investigation using other convergence tests like the limit comparison test or the root test is required.

Step-by-step solution

Apply the Ratio Test

To apply the ratio test, we need to find the limit of the absolute value of the ratio of consecutive terms: $$\lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right|$$ First, let's find the expression for \(a_{n+1}\): $$a_{n+1} =\frac{(n+1)^{100}}{(n+2)!}$$

Compute the ratio

Now, we will compute the ratio of consecutive terms: $$\frac{a_{n+1}}{a_n} = \frac{\frac{(n+1)^{100}}{(n+2)!}}{\frac{n^{100}}{(n+1)!}}$$ Simplify the expression: $$\frac{a_{n+1}}{a_n} = \frac{(n+1)^{100}(n+1)!}{(n^{100})(n+2)!}$$ ρ(n)= \frac{(n+1)^{100}(n) !}{(n+2) !(n)^{100}}$$ ρ(n)= \frac{(n+1)^{100}}{(n+2)(n+1)}$$

Compute the limit

Now, we will compute the limit as n approaches infinity: $$\lim_{n \to \infty} \left|\frac{(n+1)^{100}}{(n+2)(n+1)}\right|$$ Divide the numerator and the denominator by \((n+1)^{100}\) $$\lim_{n \to \infty} \left|\frac{1}{\left(\frac{n+2}{n+1}\right)}\right|$$ As n approaches infinity, the expression inside the limit becomes: $$\lim_{n \to \infty} \left|\frac{1}{\frac{n}{n+1}+1}\right| = 1$$

Analyze the result

Since the limit result from the ratio test is 1, the test is inconclusive. The ratio test does not give us enough information to determine if the series converges or not. Although the ratio test was inconclusive, it's important to try other tests when the initial attempt is inconclusive. We would suggest exploring other convergence tests like the limit comparison test or the root test to determine if the given series converges or not.

Key concepts

Ratio Test

The Ratio Test is a powerful tool to determine the convergence or divergence of an infinite series. It evaluates the limit of the absolute ratio of consecutive terms in a series.
The idea behind the test is simple:

• If the limit is less than 1, the series converges absolutely.
• If the limit is greater than 1, the series diverges.
• If the limit equals 1, the test is inconclusive and other methods must be used.

In the given problem, we applied the Ratio Test to the series \( \sum_{k=1}^{\infty} \frac{k^{100}}{(k+1) !}\).
We calculated the ratio of consecutive terms and took the limit as n approaches infinity, which resulted in a limit of 1. Unfortunately, a limit of 1 means the Ratio Test is inconclusive in determining whether the series converges or diverges.

Convergence Tests

Convergence Tests are a set of methods used to analyze whether an infinite series will converge to a finite value or not. These methods help in understanding the behavior of series by applying different strategies:

• Comparison Test
• Limit Comparison Test
• Ratio Test
• Root Test
• Integral Test

Each of these tests has its specific conditions and applicability. If one test is inconclusive, such as the Ratio Test in our scenario, other tests like the Limit Comparison Test or the Root Test might prove useful.
It's essential to have a repertoire of these tests, especially since some are more suited for series with particular formats or growth patterns.

Limit of a Sequence

The Limit of a Sequence refers to the value that the terms of a sequence approach as the index goes to infinity.
For sequences where each term is successive, we write this formally as \( \lim_{n \to \infty} a_n = L \). If such a limit exists, the sequence is said to converge to \( L \).
The behavior of an infinite series, particularly its convergence or divergence, is closely linked to the limit of its sequence of partial sums.
In our previous application of the Ratio Test, the evaluation was essentially looking at the behavior of each term in relation to the next, indicative of its tendency to settle around a value or drift off to infinity.

Limit Comparison Test

The Limit Comparison Test is useful when dealing with series that resemble well-known benchmark series in terms of growth behavior. It compares the given series with another that never strays beyond reasonable bounds.
The way it works:

• If you have two series \( \sum a_n \) and \( \sum b_n \), compute \( \lim_{n \to \infty} \frac{a_n}{b_n} = c \).
• If \( c \) is a positive, finite number, then both series either converge or diverge together.

Applying this test can be especially helpful when simple series like harmonic or geometric are involved, as the behavior of these series is well-understood. In cases where the Ratio Test fails to provide an answer, like our series, the Limit Comparison Test could be the next step to take.