Question

Find the first few terms of the Maclaurin series for each of the following functions and check your results by computer. $$\ln \left(1+x e^{x}\right)$$

Short answer

The first few terms are f(x) = ... . They match the computed series.

Step-by-step solution

Understand the Maclaurin Series

The Maclaurin series is a special case of the Taylor series centered at zero. It expands a function into a sum of terms that are derivatives of that function at zero: ... ].

Find the first derivative

Compute the first derivative of the function. Let \(f(x) = \ln(1 + x e^x)\). To find \(f'(x)\), use the chain rule: [ ... ]

Calculate higher-order derivatives

Find the second and third derivatives: f''(x) = \displaystyle \frac{d}{dx} f'(x) = ( ... f'''(x) = \displaystyle ( ... Terms evaluation at \(x=0\) for the first few fits

Evaluate the derivatives at x = 0

Compute the derivatives at \(x = 0\): . Therefore,

Form the Maclaurin series

Using the evaluated derivatives, collect the first few terms of the series: f(x) = ... First few terms of the Maclaurin series:

Verify using a computer

Use computational tools like Python/Mathematica/WolframAlpha to find the series. ... The computed series matches the analytically determined one.

Key concepts

Maclaurin Series

The Maclaurin series is a special kind of Taylor series centered at zero. For any function, the Maclaurin series expresses it as an infinite sum of its derivatives evaluated at zero.
The general form of a Maclaurin series for a function \(f(x)\) is:
\[ f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \text{...} \]
Each term includes a derivative of the function at zero and a factor of \(x\) raised to the power of the term index. You obtain these derivatives step by step and plug them into this formula.
For instance, in this exercise, if we calculate the Maclaurin series for \( \text{ln}(1 + x e^x)\), you would:
1. Define \(f(x)\) as \(\text{ln}(1 + x e^x)\)
2. Compute the derivatives \(f'(x)\), \(f''(x)\), \(f'''(x)\) at \(x = 0\)
3. Substitute these derivatives into the Maclaurin series formula.

Taylor Series

The Taylor series generalizes the Maclaurin series to any point, not just zero.
The Taylor series for a function \(f(x)\) around a point \(a\) is:
\[ f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \text{...} \]
Like the Maclaurin series, the Taylor series converts a function into an infinite sum of terms, each involving derivatives. However, the center point \(a\) provides flexibility in approximating the function around different points. In practice, you often truncate these infinite series after a few terms for simplicity, leading to polynomial approximations of the function. Imagine expanding \(\text{ln}(1 + x e^x)\) at different points to see how the series form changes.

Derivatives

Derivatives are essential for forming both Taylor and Maclaurin series. A derivative of a function gives us the slope or rate of change at a point.
For a function \(f(x)\), the first derivative \(f'(x)\) describes the slope, the second derivative \(f''(x)\) describes the concavity, and so on. In our example with \(\text{ln}(1 + x e^x)\), derivatives are crucial:
- First derivative \(f'(x)\): Using the chain rule, you find the rate of change at any \(x\)
- Higher-order derivatives \(f''(x)\), \(f'''(x)\): These describe deeper levels of changes and are obtained by differentiating repeatedly.
Evaluating all derivatives at \(x=0\) lets us write out the Maclaurin series.

Computational Tools

In modern mathematics, computational tools play a huge role in simplifying and verifying derivatives and series.
Tools like Python, Mathematica, or WolframAlpha can automatically compute derivatives, series expansions, and more. Python offers libraries like SymPy to symbolically compute the Maclaurin series with ease.
Here's an example in Python:
```python
import sympy as sp
x = sp.symbols('x')
f = sp.ln(1 + x * sp.exp(x))
series = sp.series(f, x, 0, 4)
print(series)
```
This level of automation ensures accuracy and saves time, making it easier for students to learn and cross-check their manual solutions.