Integration of Power and Taylor Series

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 $x=0$.