Integration of Power and Taylor Series

What does the antiderivative of the power series $f(x) = extstyle oldsymbol{igg(rac{C_n(x-a)^n}{n+1}}igg) + C$ represent?

In a Taylor series, how is the coefficient $a_n$ calculated?

Which statement is true for the radius of convergence $R$ for a power series?

What is the Maclaurin series for a function $f(x)$ centered at $0$?

If $f(x)$ is a power series, which of the following derivatives $f^{(n)}(0)$ gives the coefficient for $x^n$?

Integration of Power Series

If a power series $f(x) = \sum_{n=0}^{\infty} C_n(x-a)^n$ has a radius of convergence $R>0$, then its antiderivative can be found by integrating term by term:
- $\int f(x) dx = [\sum_{n=0}^{\infty} \frac{C_n(x-a)^{n+1}}{n+1}] + C$
- This formula holds for $x \in (a-R, a+R)$.

Taylor and Maclaurin Series

A power series representation of a function $f(x) = \sum_{n=0}^{\infty} a_nx^n$ with radius of convergence $R>0$ can be used to express the function as an infinite sum of terms.
The coefficients $a_n$ are determined by the derivatives of the function at $x=0$.
To find the coefficients:
- Compute the derivatives of $f(x)$ up to the desired order.
- Evaluate each derivative at $x=0$.
- The coefficient $a_n$ is equal to the $n$th derivative of $f(x)$ evaluated at $x=0$ divided by $n!$.
The Maclaurin Series is a specific case of the Taylor Series centered at $x=0$.
- The formula for the Maclaurin Series is:
  - $\sum_{n=0}^{\infty} \frac{f^n(0)}{n!} x^n = f(0) + f'(0)x + \frac{f''(0)}{2}x^2 + \frac{f'''(0)}{6}x^3 + \frac{f''''(0)}{24}x^4 + ...$
- It represents the function $f(x)$ as an infinite sum of terms involving the derivatives of the function evaluated at $x=0$.

Explore the integration of power series and the derivation of Taylor and Maclaurin series. This quiz covers the formulas for antiderivatives, radius of convergence, and the computation of coefficients for series representation. Test your understanding of how series can express functions as infinite sums.