Taylor series


b. Write the power series using summation notation.


f(x) = 2ˣ, a = 1

Verified step by step guidance

1. Recall the Taylor series formula for a function \( f(x) \) centered at \( a \): \[ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n \] where \( f^{(n)}(a) \) is the \( n \)-th derivative of \( f \) evaluated at \( a \).
2. Identify the function and the center: here, \( f(x) = 2^x \) and \( a = 1 \). We need to find the derivatives of \( f(x) \) evaluated at \( x = 1 \).
3. Calculate the \( n \)-th derivative of \( f(x) = 2^x \). Since \( 2^x = e^{x \ln 2} \), the derivative is: \[ f^{(n)}(x) = (\ln 2)^n 2^x \] Therefore, at \( x = 1 \), \[ f^{(n)}(1) = (\ln 2)^n 2^1 = (\ln 2)^n \cdot 2 \].
4. Substitute \( f^{(n)}(1) \) into the Taylor series formula: \[ f(x) = \sum_{n=0}^{\infty} \frac{(\ln 2)^n \cdot 2}{n!} (x - 1)^n \].
5. Write the final power series in summation notation: \[ f(x) = 2 \sum_{n=0}^{\infty} \frac{(\ln 2)^n}{n!} (x - 1)^n \].

Key Concepts

Taylor Series Expansion

A Taylor series represents a function as an infinite sum of terms calculated from the derivatives of the function at a single point. For a function f(x) centered at a point a, it is expressed as f(x) = Σ [f⁽ⁿ⁾(a)/n!] * (x - a)ⁿ, where n! is factorial and f⁽ⁿ⁾(a) is the nth derivative at a.

Derivatives of Exponential Functions

The function f(x) = 2ˣ is an exponential function whose derivatives follow a specific pattern. Each derivative is proportional to 2ˣ multiplied by the natural logarithm of 2 raised to the power of the derivative order, i.e., f⁽ⁿ⁾(x) = (ln 2)ⁿ * 2ˣ. This pattern is essential for finding terms in the Taylor series.

Summation Notation for Power Series

Summation notation (Σ) concisely expresses infinite sums, such as power series. Writing a Taylor series in summation form involves identifying the general term and using Σ from n=0 to infinity, which simplifies representation and calculation of series expansions.