Question

Find the Taylor series centered at \(x=a\) and its corresponding radius of convergence for the given function. In most cases, you need not employ the direct method of computation of the Taylor coefficients. a. \(f(x)=\sinh x, a=0\) b. \(f(x)=\sqrt{1+x}, a=0\). c. \(f(x)=\ln \frac{1+x}{1-x}, a=0\) d. \(f(x)=x e^{x}, a=1\). e. \(f(x)=\frac{1}{\sqrt{x}}, a=1\) f. \(f(x)=x^{4}+x-2, a=2\) g. \(f(x)=\frac{x-1}{2+x}, a=1\)

Short answer

a. Taylor series is \(sinh(x)=x+\frac{x^3}{3!}+\frac{x^5}{5!}+...\) and the radius of convergence is \(+\infty\). b. Taylor series is \(1+\frac{x}{2}-\frac{x^2}{8}+...\) and the radius of convergence is 1. c. Taylor series is \(2\sum_{n=0}^\infty \frac{x^{2n+1}}{2n+1}\) and the radius of convergence is 1. d. Taylor series is \(2- (x-1)+ \frac{(x-1)^2}{2!}-...\) and the radius of convergence is \(+\infty\). e. Taylor series is \(1- \frac{1}{2}(x-1) + \frac{3}{8}(x-1)^2 -...\) and the radius of convergence is 1. f. Taylor series is \(x^{4}+x-2\) and the radius of convergence is \(+\infty\). g. Taylor series is \(1-3 +\frac{3x}{2} - \frac{3x^2}{4}+...\) and the radius of convergence is 2.

Step-by-step solution

Problem a. \(f(x) = sinh(x), a=0\)

The sinh function expands directly into a Taylor series. \(sinh(x)=x+\frac{x^3}{3!}+\frac{x^5}{5!}+...\). It represents the function for all real numbers, so the radius of convergence is \(+\infty\).

Problem b. \(f(x) = \sqrt{1+x}, a=0\).

We can use the binomial series formula: \((1+x)^k = 1 + kx + \frac{k(k-1)x^2}{2!}+...\) If we set k=1/2, we have the Taylor series for the function. Hence, \(f(x)=\sqrt{1+x} = 1+\frac{x}{2}-\frac{x^2}{8}+...\). Radius of convergence is 1.

Problem c. \(f(x)=\ln \frac{1+x}{1-x}, a=0\).

Simplify the function as the difference of natural logarithms. \[f(x)=\ln(1+x)-\ln(1-x)=2\sum_{n=0}^\infty \frac{x^{2n+1}}{2n+1}\]. With this representation, the radius of convergence is 1.

Problem d. \(f(x)=xe^{x}, a=1\).

Here we can use the known Taylor Series for \(e^x\) which is \(e^x = \sum_{n=0}^{+\infty} \frac{x^n}{n!}\). For \(f(x)=xe^{x}\), shift x to (x-1) considering a=1 and apply the above Taylor series. Hence, \(f(x)= (x-1)e^{x-1} = 2- (x-1)+ \frac{(x-1)^2}{2!}-...\). The radius of convergence is \(+\infty\).

Problem e. \(f(x)=\frac{1}{\sqrt{x}}, a=1\).

Here also apply the binomial series expansion with k=-1/2. Hence, \(\frac{1}{\sqrt{x}}=(1-(x-1))^{-1/2} = 1- \frac{1}{2}(x-1) + \frac{3}{8}(x-1)^2 -...\). Radius of convergence is 1.

Problem f. \(f(x)=x^{4}+x-2, a=2\).

This is a polynomial and hence its Taylor series is itself. So, \(f(x)=x^{4}+x-2\). The radius of convergence is \(+\infty\).

Problem g. \(f(x)=\frac{x-1}{2+x}, a=1\).

Here, simplify the function using partial fractions. This will result in a geometric series. So, \(f(x)=\frac{x-1}{2+x}=1-\frac{3}{2+x}=1-3(1-\frac{x}{2})^{-1} = 1-3 +\frac{3x}{2} - \frac{3x^2}{4}+...\). The radius of convergence is 2.

Key concepts

Radius of Convergence

In calculus, the radius of convergence is an essential concept to understand when working with power series. It determines the interval in which a series converges to a function. Imagine it as measuring how far out you can go from the center and still find the function accurately represented by the series.

The radius of convergence is found using the ratio test or the root test, giving us a real number or extending to infinity.

• If the radius is infinite, like in the case of \(\sinh(x)\), the series converges everywhere.
• If the radius is 1, it means the series converges only within 1 unit of the center, as seen with the function \(\sqrt{1+x}\).
• Convergence is limited to within the radius, ensuring accurate representation of the function.

This concept is crucial because it tells you where the Taylor series can represent the function exactly. Outside this boundary, the series might diverge or provide an inaccurate approximation.

Sinh Function Expansion

The Taylor series for the hyperbolic sine function, \(\sinh(x)\), is beautifully simple. If you're familiar with the sine function from trigonometry, the hyperbolic version shares similarities but applies to hyperbolic geometry.

The expansion is given by: \[ \sinh(x) = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \ldots \]
This series contains only odd powers of \(x\). Each term becomes smaller, multiplied by the factorial in the denominator, leading to rapid convergence.

• The series converges for all real numbers, which means its radius of convergence is infinite.
• It's useful in physics and engineering, especially with hyperbolic functions that model things like catenary curves.

Understanding the expansion of \(\sinh(x)\) can deepen your comprehension of hyperbolic functions, revealing a close link to exponential functions.

Binomial Series

The binomial series is a versatile tool in mathematics, especially when dealing with functions like \(\sqrt{1+x}\). This series generalizes the binomial theorem to non-integer exponents, giving power to solve a range of complex problems.

For \(\sqrt{1+x}\), we use the binomial series formula: \[ (1+x)^k = 1 + kx + \frac{k(k-1)x^2}{2!} + \ldots \]Setting \(k=\frac{1}{2}\), this becomes: \[ \sqrt{1+x} = 1 + \frac{x}{2} - \frac{x^2}{8} + \ldots \]

• This series converges for \(-1 < x < 1\), giving a radius of convergence of 1.
• Its application ranges from algebra to calculus, facilitating the approximation of square roots and other powers.
• Understanding it helps in expanding various functions into infinite series, crucial for solving equations analytically.

The binomial series not only simplifies calculations but also underpins many mathematical transformations and expansions.