Problem 7 - The Calculus with Analytic Geometry

Question

Determine if the given series is convergent or divergent.\(\sum_{k=1}^{+\infty} \frac{n !}{(n+2) !}\)

Step-by-Step Solution

Answer

The series \(\[\begin{equation}\sum_{n=1}^{+\infty} \operatorname{csch}(n)\end{equation}\]\) is convergent.

Step 1: Understand the Series

The given series is \(\[\begin{equation}\sum_{n=1}^{+\infty} \operatorname{csch}(n)\end{equation}\]\). The term \(\[\begin{equation}\operatorname{csch}(n)\end{equation}\]\) represents the hyperbolic cosecant function, defined as \(\[\begin{equation}\operatorname{csch}(x) = \frac{1}{\sinh(x)}\end{equation}\]\).

Step 2: Examine the Behavior of the Terms

Consider the behavior of \(\[\begin{equation}\operatorname{csch}(n)\end{equation}\]\) as \(\[\begin{equation}n\end{equation}\]\) goes to infinity. The hyperbolic sine function \(\[\begin{equation}\sinh(x)\end{equation}\]\) is defined as \(\[\begin{equation}\sinh(x) = \frac{e^x - e^{-x}}{2}\end{equation}\]\). For large values of \(\[\begin{equation}x\end{equation}\]\), \(\[\begin{equation}sinh(x)\end{equation}\]\) approximately equals \(\[\begin{equation}\frac{e^x}{2}\end{equation}\]\).

Step 3: Simplify the Series Terms

So, for large \(\[\begin{equation}n\end{equation}\]\), \(\[\begin{equation}\operatorname{csch}(n)\end{equation}\]\) is approximated by \(\[\begin{equation}\frac{2}{e^n}\end{equation}\]\). Hence, \(\[\begin{equation}\operatorname{csch}(n)\end{equation}\]\) behaves similarly to a geometric sequence with a common ratio of \(\[\begin{equation}\frac{1}{e}\end{equation}\]\), which is less than 1.

Step 4: Determine Convergence

Since the terms \(\[\begin{equation}\operatorname{csch}(n)\end{equation}\]\) are equivalent to \(\[\begin{equation}\frac{2}{e^n}\end{equation}\]\) for large \(\[\begin{equation}n\end{equation}\]\), the series \(\[\begin{equation}\sum_{n=1}^{+\infty}\frac{2}{e^n}\end{equation}\]\) is a convergent geometric series. Thus, our original series \(\[\begin{equation}\sum_{n=1}^{+\infty} \operatorname{csch}(n)\end{equation}\]\) is also convergent.

Key Concepts

hyperbolic functions

Hyperbolic functions are analogs of trigonometric functions but for a hyperbola rather than a circle. The hyperbolic sine \( \sinh(x) \ \) and hyperbolic cosine \( \cosh(x) \ \) are defined as follows:
\[ \sinh(x) = \frac{e^x - e^{-x}}{2} \ \] and \[ \cosh(x) = \frac{e^x + e^{-x}}{2} \ \] .
These functions are significant in various fields such as physics, engineering, and mathematics, particularly for problems involving hyperbolic geometry, Laplace equations, and systems with hyperbolic symmetry.
The hyperbolic cosecant \( \operatorname{csch}(x) \ \) is one of the hyperbolic functions and is defined as the reciprocal of the hyperbolic sine, \[ \operatorname{csch}(x) = \frac{1}{\sinh(x)} \ \] .
This function plays a vital role in the given exercise.
Understanding how these functions behave for large values of \( x \) helps analyze the convergence properties of related series.

geometric series

A geometric series is a series of numbers in which each term after the first is found by multiplying the previous term by a constant called the common ratio.
The general form of a geometric series is: \[ a + ar + ar^2 + ar^3 + \cdots \ \]
where \( a \) is the first term and \( r \) is the common ratio.
The series converges when the absolute value of the common ratio is less than 1 \( (\mid r \mid < 1) \) , and its sum can be calculated using the formula: \[ \sum_{n=0}^{\infty} ar^n = \frac{a}{1-r} \ \] .
Let's contrast this with our problem. When examining \( \operatorname{csch}(n) \) for large \( n \), it approximates \[ \frac{2}{e^n} \ \] .
This resembles the form of a geometric series with \( a = 2 \) and \( r = \frac{1}{e} \). Since \( \frac{1}{e} \) is less than 1, the series converges.

infinite series

An infinite series is a sum of an infinite sequence of terms. Determining whether an infinite series converges or diverges is a fundamental topic in calculus.
To determine convergence:

• We can use tests like the Ratio Test, Root Test, or Comparison Test.
• If the series meets certain criteria, we can conclude whether it sums to a finite value (converges) or does not (diverges).

In the given exercise, we used the behavior of the terms of the series for large \( n \).
The terms \( \operatorname{csch}(n) \) approximating \[ \frac{2}{e^n} \ \] allowed us to recognize it as a form of the geometric series. Since \( \frac{1}{e} \) is less than 1, by properties of geometric series, we concluded that our infinite series converges.
This approach helps students understand both the series itself and the broader methods for determining series convergence.