Problem 31 - Single Variable Calculus

Question

Determine whether the sequence \(\left\\{a_{n}\right\\}\) converges or diverges. If it converges, find its limit. \(a_{n}=\tanh n\)

Step-by-Step Solution

Answer

The sequence \(a_n = \tanh n\) converges, and its limit is 1 as \(n\) goes to infinity.

Step 1: Review the definition of hyperbolic tangent

The hyperbolic tangent function \(\tanh x\) is defined as: \[\tanh x = \frac{\sinh x}{\cosh x} = \frac{e^x - e^{-x}}{e^x + e^{-x}}\]

Step 2: Analyze the behavior of \(\tanh x\) as \(x\) goes to infinity

As \(x\) goes to infinity, \(e^x\) grows much faster than \(e^{-x}\). Therefore, the terms \(e^x\) and \(-e^{-x}\) in the numerator and denominator will dominate the expression. \[\lim_{x\to\infty}\tanh x = \lim_{x\to\infty}\frac{e^x - e^{-x}}{e^x + e^{-x}} \approx \lim_{x\to\infty}\frac{e^x}{e^x} = 1\]

Step 3: Analyze the behavior of \(\tanh x\) as \(x\) goes to negative infinity

As \(x\) goes to negative infinity, \(e^{-x}\) grows much faster than \(e^x\). Therefore, the terms \(-e^x\) and \(e^{-x}\) in the numerator and denominator will dominate the expression. \[\lim_{x\to-\infty}\tanh x = \lim_{x\to-\infty}\frac{e^x - e^{-x}}{e^x + e^{-x}} \approx \lim_{x\to-\infty}\frac{-e^{-x}}{e^{-x}} = -1\]

Step 4: Determine if the sequence converges or diverges

The sequence \(a_n = \tanh n\) will converge if it approaches a finite limit as \(n\) goes to infinity. As we found in step 2, the limit of \(\tanh x\) as \(x\) goes to infinity is 1, so the sequence does converge: \[\lim_{n\to\infty} a_n = \lim_{n\to\infty}\tanh n = 1\]

Step 5: Provide the final answer

The sequence \(a_n = \tanh n\) converges, and its limit is 1 as \(n\) goes to infinity.

Key Concepts

Hyperbolic Tangent

The hyperbolic tangent function, denoted as \(\tanh x\), is a widely used function in mathematics, especially in calculus and complex analysis. It can be defined in several ways, one of which is the quotient of the hyperbolic sine function \(\sinh x\) and the hyperbolic cosine function \(\cosh x\), resulting in \(\tanh x = \frac{\sinh x}{\cosh x}\). Alternatively, using the exponentials functions, it can be expressed as \(\tanh x = \frac{e^x - e^{-x}}{e^x + e^{-x}}\).

This function exhibits certain features making it particularly interesting: it is odd, meaning \(\tanh(-x) = -\tanh(x)\), and it has horizontal asymptotes as \(x\) approaches positive or negative infinity, which are \(1\) and \(-1\), respectively. The values of \(\tanh x\) always lie between -1 and 1, making it a bounded function.

Limits of Sequences

The concept of limits is fundamental in sequence analysis and calculus. In simple terms, a limit tries to describe the value that a sequence \(\{a_n\}\) approaches as the index \(n\) becomes very large. If as \(n\) goes to infinity, the terms of the sequence approach a single, finite number, we say that the sequence converges to that number, which is known as the limit of the sequence. Conversely, if the sequence does not approach a single value, it diverges. Calculating the limit involves analyzing dominant terms, especially when dealing with indeterminate forms or sequences involving exponential growth or decay.

Exponential Functions

Exponential functions are powerful mathematical tools that model a wide array of phenomena, from compound interest to radioactive decay. An exponential function has the form \(f(x) = a^x\), where \(a\) is a positive constant base, and \(x\) is the exponent. These functions are characterized by the fact that their rate of growth (or decay, if \(0 < a < 1\)) is proportional to their current value. This property makes the exponential function increase very rapidly as \(x\) grows when \(a > 1\). In fact, for large positive values of \(x\), the exponential function with a base greater than 1 grows much faster than any polynomial function of \(x\).

Convergence of Sequences

A sequence is said to converge if its terms approach a fixed value, called the limit, as the index \(n\) goes to infinity. The convergence of the sequence depends on its terms becoming arbitrarily close to the limit for all sufficiently large indices. In the case of the hyperbolic tangent function, as \(n\) becomes increasingly large, the sequence \(a_n = \tanh n\) approaches 1, which means it converges to 1. This behavior is indicative of the stability of the function over the long term — regardless of the values of \(n\) we choose beyond a certain point, the function will yield a value very close to our limit, lying within a predictable and consistent range.