Problem 27 - Thomas Calculus

Question

Use power series operations to find the Taylor series at \(x=0\) for the functions in Exercises \(13-30\) . $$e^{x}+\frac{1}{1+x}$$

Step-by-Step Solution

Answer

The Taylor series at \( x=0 \) is \( 2 + \frac{3x^2}{2} - \frac{5x^3}{6} + \cdots \) with convergence \(|x| < 1\).

Step 1: Recall the Taylor Series of Known Functions

The Taylor series for a function at a point is an infinite sum representing the function as a power series. We know:1. The Taylor series for \( e^x \) at \( x = 0 \) is given by: \[e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots\]2. The Taylor series for \( \frac{1}{1+x} \) at \( x = 0 \) is a geometric series \(|x| < 1\):\[\frac{1}{1+x} = \sum_{n=0}^{\infty} (-1)^n x^n = 1 - x + x^2 - x^3 + \cdots\]

Step 2: Add the Two Series Together

Since the Taylor series of a sum is the sum of the Taylor series, add the series for \( e^x \) and \( \frac{1}{1+x} \):Combine corresponding terms from each series:\[(e^x) + \left( \frac{1}{1+x} \right) = (1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots ) + (1 - x + x^2 - x^3 + \cdots )\]Group and simplify:\[= (1 + 1) + (x - x) + \left(\frac{x^2}{2} + x^2\right) + \left(\frac{x^3}{6} - x^3\right) + \cdots\]

Step 3: Simplify the Resulting Series

Combine like terms:\[= 2 + 0 + \left(\frac{3x^2}{2}\right) + \left(\frac{x^3}{6} - x^3\right) + \cdots\]Simplify further:\[= 2 + \frac{3x^2}{2} - \frac{5x^3}{6} + \cdots\]Thus, the Taylor series for the given function at \( x = 0 \) is:\[2 + \frac{3x^2}{2} - \frac{5x^3}{6} + \cdots\]

Step 4: Convergence Considerations

The functions \( e^x \) and \( \frac{1}{1+x} \) converge for all real numbers and \(|x| < 1\), respectively. Therefore, the resulting series' valid convergence range is \(|x| < 1\). However, the Taylor series constructed continues to represent terms valid as far as provided.

Key Concepts

Power Series

A power series is a way to represent a function as an infinite sum of terms involving powers of a variable. These series take the form given by:

• \[ \sum_{n=0}^{\infty} a_n (x - c)^n \]

Here, \( a_n \) are the coefficients, \( x \) is the variable, and \( c \) is the center of the series. Power series can converge to represent a function over a certain range of \( x \), allowing us to use algebraic operations on infinite series of terms.
They are particularly useful because they provide a straightforward way to approximate transcendental functions like trigonometric, exponential, and logarithmic functions.
With each additional term added, the approximation improves, getting closer to the function's true value.

Geometric Series

A geometric series is a specific type of power series where each term after the first is produced by multiplying the previous term by a constant factor. The general form of a geometric series is:

• \[ a + ar + ar^2 + ar^3 + \cdots = \sum_{n=0}^{\infty} ar^n \]

Here, \( a \) is the first term, and \( r \) is the constant ratio. A geometric series converges, or sums to a finite value, given that \(|r| < 1\).
This property of convergence plays a key role when using geometric series in computations, such as deriving the Taylor series for specific functions.
An example used in our context is the Taylor series expansion of \( \frac{1}{1+x} \), which resembles a geometric series with initial term \( 1 \) and ratio \(-x\).

Convergence

Convergence refers to the behavior of a series as more terms are added. For a series to converge, its sum must approach a finite limit as the number of terms tends towards infinity. Not all series are convergent;
some can diverge, meaning their sum doesn't approach a specific value.
For Taylor and power series, determining convergence is crucial because it indicates where the series accurately represents the function it's meant to approximate.
In the case of the Taylor series for \( e^x + \frac{1}{1+x} \), convergence is conditional:

• The series for \(e^x\) converges for all real numbers because exponentials inherently converge everywhere.
• The series for \(\frac{1}{1+x}\) converges for \(|x|<1\).

Thus, the combined series also converges within this range, making its use valid for equations involving those terms.

Exponential Function

The exponential function, denoted as \( e^x \), is a mathematical function with applications in various fields such as biology, economics, and physics. The exponential function is unique because it grows at a rate proportional to its value, a property seen in natural processes like population growth. The Taylor series expansion provides a way to express \( e^x \) using power series:

• \[ e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots = \sum_{n=0}^{\infty} \frac{x^n}{n!} \]

This series converges with increasing accuracy as more terms are added. Each term involves a factorial, which grows quickly, ensuring the term sizes decrease rapidly, aiding in convergence. In practice, this series is not only theoretical but also utilized in calculations where an exact evaluation of \( e^x \) isn't possible and an approximate value through series can be valuable.