[Math] Represent the function $f(x)=x^{0.3}$ as a Taylor series centered at $5$

calculuspower seriestaylor expansion

Represent the function $f(x)=x^{0.3}$ as a power series $\sum_{n=0}^\infty c_n(x-5)^n$

Find the following coefficients: $c_0$, $c_1$, $c_2$, $c_3$

Here are my answers:

  • $c_0= 5^{0.3} $
  • $c_1= 0.3 \cdot 5^{-0.7} $
  • $c_2= -0.2 \cdot 5^{-1.7} $
  • $c_3 = 0.35 \cdot 5^{-2.7}$

What am I doing wrong? I know $c_0$ and $c_1$ are correct, but what is wrong with $c_2$ and $c_3$?

Best Answer

We find the coefficient $c_2$ of $x^2$.

We have $f''(x)=(0.3)(-0.7)x^{-1.7}$. Evaluate at $x=5$, divide by $2!$. You have two little errors, replacing $-0.21$ by $-0.2$, and forgetting to divide by $2!$.

Related Question