This week, we’ll work out some Taylor Series expansions of popular functions. • f (x) = (1−x) • f (x) = ex • f (x) = ln(1 + x) • f(x)=x(1/2) For each function, only consider its valid ranges as indicated in the notes when you are computing the Taylor Series expansion. Please submit your assignment as an R- Markdown document.
The Taylor series expansion of \(f(x) = (1-x)\) about \(x = 0\) is given by:
∞f(x)= ∑ (-1)nxn n=0
This series converges when \(|x| < 1\). Therefore, the valid range for this series is \(-1 < x < 1\).
The Taylor series expansion of \(f(x) = e^x\) about \(x = 0\) is given by: ∞ f(x)= ∑ (-1)x^n/n! n=0
This series converges for all values of \(x\). Therefore, the valid range for this series is \(-\infty < x < \infty\).
The Taylor series expansion of \(f(x) = \ln(1+x)\) about \(x = 0\) is given by: ∞ f(x)= ∑ (-1)n+1x^n/n n=1
This series converges when \(-1 < x \leq 1\). Therefore, the valid range for this series is \(-1 < x \leq 1\).
The Taylor series expansion of \(f(x) = x^{1/2}\) about \(x = 1\) is given by: ∞ f(x)= ∑ fn(1)/n!(x-1)n n=0
where \(f^{(n)}(x)\) is the \(n\)th derivative of \(f(x)\). For \(f(x) = x^{1/2}\), we have:
f’(x)= 1/2√x, f’‘(x)=1/4x^3/2, f’’’(x)=3/8x^5/2, …
Evaluating these derivatives at \(x = 1\), we have:
f’(1)=1/2, f’‘(1)-1/4, f’’’(1)=3/8, …
f(x) = {n=0}^ (x - 1)^n = {n=0}\frac{(-1){n-1} (2n-1)!!