Finding the Power Series of f(x) = 10/(1-5x)

[ganondorf29]

Homework Statement

Determine the series of the given function:

f(x) = 10 / (1-5*x)

Homework Equations

Power series of 1/(1-x) = Σ from n=0 to n=infinity of (x^n)

The Attempt at a Solution

f(x) = 10/(1-5x)
= 10*(1/1-5x)
= 10 * Σ(5x)^n
= 10 * Σ(5^n)*(x^n)
= Σ (50^n)*(x^n) <--- Not sure if that is right

Any help would be appreciated. Thank you

 

[Dick]

Not right. The series part is fine. But 10*(5^n) does not equal 50^n. Think about, say, n=2.

 

[ganondorf29]

Is it just Σ(5^n)*(x^n)*10 ?

 

[Dick]

I think that's the simplest way to write it, yes.

 

[ganondorf29]

One more thing. To find the interval on convergence, I know I have to take the ratio test as n-->infinity. Is this how I'm supposed to set it up?

lim [x^(n+1) * 5^(n+1) / (n+1)*(n+1)] * [(n*n/x^n*5^x)]
n->inf

After canceling out some factors I got:

lim 1/(2n+1) = 0
n->inf

Is that right?

 

[Dick]

No. Where are all those n+1 and n's coming from? The nth term of your series a_n=10*5^n*x^n. So the ratio of a_(n+1)/a_n is just 10*5^(n+1)*x^(n+1)/(10*5^n*x^n) isn't it? What's that?