Question

The inverse hyperbolic sine is defined in several ways; among them are $$\sinh ^{-1} x=\ln (x+\sqrt{x^{2}+1})=\int_{0}^{x} \frac{d t}{\sqrt{1+t^{2}}}$$ Find the first four terms of the Taylor series for \(\sinh ^{-1} x\) using these two definitions (and be sure they agree).

Short answer

Answer: The first four terms of the Taylor series for sinh^-1(x) using both definitions are 1, 0, -1/6, and 0.

Step-by-step solution

Taylor series expansion of natural logarithm definition

The given definition of the inverse hyperbolic sine is: ``` sinh^-1(x) = ln(x + sqrt(x^2 + 1)) ``` To find the first four terms of the Taylor series, we need to find its derivatives and evaluate them at 0. Let's find the first four derivatives of sinh^-1(x), First Derivative: ``` d [sinh^-1(x)] / d(x) = 1 / (sqrt(x^2 + 1)) ``` Second Derivative: ``` d² [sinh^-1(x)] / d(x)² = (-x) / ((x^2 + 1)^(3/2)) ``` Third Derivative: ``` d³ [sinh^-1(x)] / d(x)³ = (2x^2 - 1) / ((x^2 + 1)^(5/2)) ``` Now, we need to evaluate them at x = 0. First Derivative (x = 0): ``` 1 / (sqrt(0 + 1)) = 1 ``` Second Derivative (x = 0): ``` (-0) / ((0 + 1)^(3/2)) = 0 ``` Third Derivative (x = 0): ``` (2(0)^2 - 1) / ((0 + 1)^(5/2)) = -1 ``` Now, we can write the Taylor series expansion for sinh^-1(x): ``` sinh^-1(x) = x - (1/6)x^3 + O(x^5) ``` So, the first four terms are: 1, 0, -1/6, and 0.

Taylor series expansion of integral definition

The integral definition of the inverse hyperbolic sine is: ``` sinh^-1(x) = ∫ dt/(sqrt(1 + t^2)) for t = 0 to x ``` First, expand the integrand in terms of a Taylor series: ``` (1 / sqrt(1 + t^2)) = 1 - (1/2)t^2 + (3/8)t^4 - ... ``` Now, integrate term-wise within the range of 0 to x: ``` ∫ dt/(sqrt(1 + t^2)) for t = 0 to x = t - (1/6)t^3 + (3/40)t^5 - ... ``` Now, evaluate this at x to get the Taylor series expansion for sinh^-1(x): ``` sinh^-1(x) = x - (1/6)x^3 + O(x^5) ``` So, the first four terms are: 1, 0, -1/6, and 0.

Comparing the Taylor series expansions

Now let's compare the Taylor series expansions obtained from both definitions. For the natural logarithm definition, the first four terms are: 1, 0, -1/6, and 0. For the integral definition, the first four terms are: 1, 0, -1/6, and 0. Both Taylor series expansions agree, as the first four terms are the same for both definitions. Therefore, we have found the first four terms of the Taylor series for the sinh^-1(x) function using both definitions as: 1, 0, -1/6, and 0.

Key concepts

Taylor series

The Taylor series is a powerful tool in mathematics, providing a way to approximate complex functions with polynomials. The essence of the Taylor series is to express a function as an infinite sum of terms calculated from the values of the function's derivatives at a single point.
It can be used to approximate functions that are otherwise difficult to compute.

• Each term in the Taylor series is derived from the corresponding derivative of the function at a specific point.
• The series can be truncated to give a polynomial approximation that is useful at small values around that specific point.

For the inverse hyperbolic sine \( \sinh^{-1} x \), it's insightful to find its Taylor series expansion centered at 0 and looking at the series form:

\[ \sinh^{-1}(x) = x - \frac{x^3}{6} + O(x^5) \]This expression involves the first few non-zero terms of the series, giving a close approximation when \(x\) is near 0.

natural logarithm

The natural logarithm, denoted as \( \ln(x) \), is closely tied to exponential growth and is an essential component in calculus. It is the inverse function of the exponential function with base \(e\).
When dealing with the inverse hyperbolic sine, one of the definitions presented is:
\[ \sinh^{-1}(x) = \ln(x + \sqrt{x^2 + 1}) \]This expression uses the properties of logarithms and square roots, allowing us to transform a hyperbolic function into a form amenable to Taylor series expansion. To find the series, you compute the derivatives of the logarithmic expression, evaluating them at \(x = 0\), which gives us insights into the behavior of the function near the origin.
Logarithmic differentiation is crucial here because it simplifies the process of finding higher-order derivatives needed for the Taylor series.

integration

Integration is the process of finding the integral of a function, which is the reverse operation of differentiation and is used to find areas under curves. In the context of the inverse hyperbolic sine, integration is another approach to defining the function:
\[ \sinh^{-1}(x) = \int_0^x \frac{1}{\sqrt{1 + t^2}} \, dt \]
To derive a Taylor series from this, the integrand \( \frac{1}{\sqrt{1+t^2}} \) is expanded into a series, focusing on terms that simplify the integration process.

• This process results in a series of integrals for each term.
• The integration is then performed term-by-term, accumulating to provide approximations of the function.

By evaluating the series from 0 to \(x\), we obtain the same Taylor series as from the logarithmic definition:
\[ \sinh^{-1}(x) = x - \frac{x^3}{6} + O(x^5) \] Both integration and differentiation, when applied correctly, produce equivalent Taylor series, confirming their consistent utility in calculus.

derivatives

Derivatives measure the rate of change of a function relative to changes in its input, and they are foundational to calculus. For finding the Taylor series of the inverse hyperbolic sine \( \sinh^{-1}(x) \), derivatives play a crucial role.
To derive the Taylor series from the natural logarithm definition, compute several derivatives. Each derivative represents the function's behavior at increasingly higher orders.
For instance, the first few derivatives of \( \sinh^{-1}(x) = \ln(x + \sqrt{x^2 + 1}) \) are:

• The first derivative: \( \frac{d}{dx}[\ln(x + \sqrt{x^2 + 1})] = \frac{1}{\sqrt{x^2 + 1}} \)
• The second derivative: \( \frac{-x}{(x^2 + 1)^{3/2}} \)
• The third derivative: \( \frac{2x^2 - 1}{(x^2 + 1)^{5/2}} \)

Evaluating these at \(x = 0\), we gather coefficients for the Taylor series. This method highlights how differentiation systematically builds up the series, allowing us to represent functions as polynomials valid at small input values, such as around \(x = 0\).