Find the Taylor series for f(x) centered at the given value...

Find the Taylor series for $f(x)$ centered at the given value of $a$. $f(x) = \frac{10}{x}$, $a = -4$ $\sum_{n=0}^{8} (-\frac{10}{(4)^{n+1}}(x - 4)^n)$ Find the associated radius of convergence $R$. $R = 4$

Transcript

00:01 So let's find the taylor expansion of f of x is equal to 4 over x, which is equal to 4 times 1 over x.

00:14 So when you do the taylor expansion, let's do it for example for 1 over x, which is going to be x to the n.

00:19 So 4 times x to the n, or negative n, excuse me.

00:24 I guess you could say negative 1.

00:29 Okay, so let's take a zero derivative.

00:34 It's going to be equal to 4x to the negative first.

00:37 First derivative is going to be equal to negative 4x to the negative second power.

00:47 Second derivative will be equal to negative, or positive 8x to the negative third power.

00:56 Third one will be negative 24x to the negative fourth power.

01:05 Now if we plug in something like, well in this case, let's say that we want to find this at the value a equals negative 2.

01:23 And so when a equals negative 2 for these, we'll just get, well we won't plug in the x values for now.

01:32 These are our derivatives.

01:33 Oops, i forgot to write this as f.

01:35 Excuse me.

01:36 So the taylor polynomial for 1 over x, i don't know how she'll write this.

01:44 I'll say 1 over x can be approximated as 1 minus x minus 1 plus x minus 1 squared and so on.

02:01 Notice i'm having this start at x, at a equals 1 because we can't have a, when we plug in the value, we can't plug in 0.

02:08 So we'll need to start this at 1.

02:13 We need to have this about 1...

Question

Find the Taylor series for $f(x)$ centered at the given value of $a$. $f(x) = \frac{10}{x}$, $a = -4$ $\sum_{n=0}^{8} (-\frac{10}{(4)^{n+1}}(x - 4)^n)$ Find the associated radius of convergence $R$. $R = 4$

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