Certainly! Let's take a closer look at the integration step:

Given the Taylor series expansion:
\[ e^{-x^2} = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{n!} \]

We want to integrate:
\[ \int_{0}^{1} e^{-x^2} \, dx = \int_{0}^{1} \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{n!} \, dx \]

Since the series converges uniformly on \([0, 1]\), we can interchange the sum and the integral:
\[ \int_{0}^{1} \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{n!} \, dx = \sum_{n=0}^{\infty} \int_{0}^{1} \frac{(-1)^n x^{2n}}{n!} \, dx \]

Now, let's integrate each term individually:
\[ \int_{0}^{1} \frac{(-1)^n x^{2n}}{n!} \, dx \]

This simplifies to:
\[ \frac{(-1)^n}{n!} \int_{0}^{1} x^{2n} \, dx \]

We need to evaluate the integral:
\[ \int_{0}^{1} x^{2n} \, dx \]

To integrate \( x^{2n} \):
\[ \int x^{2n} \, dx = \frac{x^{2n+1}}{2n+1} + C \]

Evaluating this definite integral from 0 to 1:
\[ \int_{0}^{1} x^{2n} \, dx = \left[ \frac{x^{2n+1}}{2n+1} \right]_{0}^{1} = \frac{1^{2n+1}}{2n+1} - \frac{0^{2n+1}}{2n+1} = \frac{1}{2n+1} \]

Thus:
\[ \int_{0}^{1} x^{2n} \, dx = \frac{1}{2n+1} \]

Substituting this back into our earlier expression:
\[ \int_{0}^{1} \frac{(-1)^n x^{2n}}{n!} \, dx = \frac{(-1)^n}{n!} \cdot \frac{1}{2n+1} \]

Now, summing over all terms \(n\) from 0 to \(\infty\):
\[ \int_{0}^{1} e^{-x^2} \, dx = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)n!} \]

Therefore, the integral \( \int_{0}^{1} e^{-x^2} \, dx \) written as an infinite series is:
\[ \int_{0}^{1} e^{-x^2} \, dx = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)n!} \]

Do you have any more questions or need further clarification?

Here are some additional questions you might consider:
1. What are some common methods for approximating definite integrals?
2. Can you provide more examples of integrating series term-by-term?
3. How do you test for uniform convergence of a series?
4. What is the significance of the factorial in the Taylor series?
5. Can you explain how the exponential function \(e^{-x^2}\) is used in probability theory?

**Tip:** When working with series and integrals, carefully consider the conditions under which you can interchange summation and integration to ensure the steps are valid.

Explore the step-by-step integration of the Taylor series expansion for the function e^(-x^2) over the interval [0, 1]. Learn how to interchange summation and integration, evaluate individual terms, and derive the infinite series representation. Suitable for advanced undergraduate students and learners of integral calculus.

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