Question

Determine convergence or divergence, with \(r>0\). (a) \(\sum \frac{n !}{r^{n}}\) (b) \(\sum n^{p} r^{n}\) (c) \(\sum \frac{r^{n}}{n !}\) (d) \(\sum \frac{r^{2 n+1}}{(2 n+1) !}\) (e) \(\sum \frac{r^{2 n}}{(2 n) !}\)

Short answer

Question: Determine the convergence or divergence of the following series: (a) \(\sum \frac{n !}{r^{n}}\) (b) \(\sum n^{p} r^{n}\) (c) \(\sum \frac{r^{n}}{n !}\) (d) \(\sum \frac{r^{2 n+1}}{(2 n+1) !}\) (e) \(\sum \frac{r^{2 n}}{(2 n) !}\) Answer: (a) The series \(\sum \frac{n !}{r^{n}}\) diverges. (b) The series \(\sum n^{p} r^{n}\) converges if \(r < 1\) and diverges if \(r > 1\). If \(r = 1\), the Ratio Test is inconclusive. (c) The series \(\sum \frac{r^{n}}{n !}\) converges. (d) The series \(\sum \frac{r^{2 n+1}}{(2 n+1) !}\) converges. (e) The series \(\sum \frac{r^{2 n}}{(2 n) !}\) converges.

Step-by-step solution

Identify the general term of the series

We have \(a_n = \frac{n !}{r^{n}}\).

Find the limit of the ratio of consecutive terms

Calculate \(\lim_{n \to \infty} \frac{a_{n+1}}{a_{n}} = \lim_{n \to \infty} \frac{(n+1) !}{r^{n+1}} \cdot \frac{r^n}{n!} = \lim_{n \to \infty} \frac{(n+1)}{r}\). Since \(r>0\), the limit is equal to \(\infty\), which is greater than 1.

Apply the Ratio Test

Since the limit of the ratio of consecutive terms is greater than 1, the series \(\sum \frac{n !}{r^{n}}\) diverges. (b) \(\sum n^{p} r^{n}\)

Identify the general term of the series

We have \(a_n = n^{p} r^{n}\).

Find the limit of the ratio of consecutive terms

Calculate \(\lim_{n \to \infty} \frac{a_{n+1}}{a_{n}} = \lim_{n \to \infty} \frac{(n+1)^p r^{n+1}}{n^p r^n} = r \lim_{n \to \infty} \left(\frac{n+1}{n}\right)^p\). The limit equals \(r\), since \(\lim_{n\to\infty}\left(\frac{n+1}{n}\right)^p=1\).

Apply the Ratio Test

The series converges if \(r < 1\) and diverges if \(r > 1\). If \(r = 1\), the Ratio Test is inconclusive. (c) \(\sum \frac{r^{n}}{n !}\)

Identify the general term of the series

We have \(a_n = \frac{r^n}{n!}\).

Find the limit of the ratio of consecutive terms

Calculate \(\lim_{n \to \infty} \frac{a_{n+1}}{a_{n}} = \lim_{n \to \infty} \frac{r^{n+1}}{(n+1)!} \cdot \frac{n!}{r^n} = \lim_{n \to \infty} \frac{r}{n+1}\). We have that the limit equals \(0\), which is less than 1.

Apply the Ratio Test

Since the limit of the ratio of consecutive terms is less than 1, the series \(\sum \frac{r^{n}}{n !}\) converges. (d) \(\sum \frac{r^{2 n+1}}{(2 n+1) !}\)

Identify the general term of the series

We have \(a_n = \frac{r^{2 n+1}}{(2 n+1) !}\).

Find the limit of the ratio of consecutive terms

Calculate \(\lim_{n \to \infty} \frac{a_{n+1}}{a_{n}} = \lim_{n \to \infty} \frac{r^{2 (n+1)+1}}{(2(n+1)+1)!} \cdot \frac{(2n+1)!}{r^{2 n+1}} = \lim_{n \to \infty} \frac{r^2}{(2n+3)(2n+2)}\). The limit equals \(0\), which is less than 1.

Apply the Ratio Test

Since the limit of the ratio of consecutive terms is less than 1, the series \(\sum \frac{r^{2 n+1}}{(2 n+1) !}\) converges. (e) \(\sum \frac{r^{2 n}}{(2 n) !}\)

Identify the general term of the series

We have \(a_n = \frac{r^{2n}}{(2n)!}\).

Find the limit of the ratio of consecutive terms

Calculate \(\lim_{n \to \infty} \frac{a_{n+1}}{a_{n}} = \lim_{n \to \infty} \frac{r^{2(n+1)}}{(2(n+1))!} \cdot \frac{(2n)!}{r^{2n}} = \lim_{n \to \infty} \frac{r^{2}}{(2n+2)(2n+1)}\). Again, the limit equals \(0\) which is less than 1.

Apply the Ratio Test

Since the limit of the ratio of consecutive terms is less than 1, the series \(\sum \frac{r^{2 n}}{(2 n) !}\) converges.

Key concepts

Ratio Test

The Ratio Test is a popular method in mathematics to determine if a series converges or diverges. To apply the Ratio Test, you focus on the limit of the ratio of every consecutive term in the series. Here's how it generally works:

• Express the general term of the series as \(a_n\).
• Calculate the ratio of the consecutive terms \(\frac{a_{n+1}}{a_n}\).
• Find the limit: \(\lim_{n \to \infty} \frac{a_{n+1}}{a_n}\).

If this limit:

• Is less than 1, the series converges.
• Is greater than 1, the series diverges.
• Equals 1, the test is inconclusive.

An example from our exercise is the series \( \sum \frac{r^n}{n!} \), where applying the Ratio Test showed a limit less than 1, indicating convergence.

Factorials in Series

Factorials, denoted by \(!\), are products of all positive integers up to a certain number. They often appear in series involving rapid growth or calculations of permutations. When factorials are in a series, they significantly affect the rate at which terms either grow or shrink.In our exercise, factorials appear in the series \( \sum \frac{n!}{r^n} \). Here, \(n!\) represents a rapid increase compared to \(r^n\), a feature which often indicates whether the series is divergent. Factorials in the denominator, as in \( \sum \frac{r^n}{n!} \), help a series shrink faster, leaning towards convergence. This crucial balance dictated by factorials is a common theme in evaluating series.

Exponential Growth in Series

Exponential growth occurs when something increases by a constant proportion over equal intervals. In mathematics, you'll often see this in terms like \(r^n\) within series. Whether a series containing exponential growth converges or diverges depends greatly on how fast the growth is compared to other parts of the series.Consider the series \( \sum n^p r^n \) from our exercise. The term \(r^n\) can grow extremely fast, often overwhelming polynomial factors like \(n^p\), leading to divergence. However, when coupled with factors growing more slowly, as in \( \sum \frac{r^n}{n!} \), exponential growth can lead to convergence because the denominator \(n!\) grows much faster than \(r^n\) can increase.

Power Series

A power series is a series of the form \( \sum a_n x^n \), where \(x\) is a variable and \(a_n\) represents coefficients. Power series are important in mathematics for representing functions as sums of infinite terms. They converge within a certain range of \(x\), known as the radius of convergence.An example is the series \( \sum \frac{r^{2n}}{(2n)!} \), where we see \(r^{2n}\), following a power series format. These concepts are not only theoretical; they also have practical applications, such as solving differential equations or approximating functions. By understanding power series, one gains insights into how functions behave and how they can be manipulated and analyzed.