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

In summary, the given function can be represented as a power series of 10*5^n*x^n. The interval of convergence can be found by taking the ratio of a_n+1 and a_n, which simplifies to 10*5^(n+1)*x^(n+1)/(10*5^n*x^n).
  • #1
ganondorf29
54
0

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
 
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  • #2
Not right. The series part is fine. But 10*(5^n) does not equal 50^n. Think about, say, n=2.
 
  • #3
Is it just Σ(5^n)*(x^n)*10 ?
 
  • #4
I think that's the simplest way to write it, yes.
 
  • #5
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?
 
  • #6
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?
 

What is the power series of f(x) = 10/(1-5x)?

The power series of f(x) = 10/(1-5x) is 10 + 50x + 250x^2 + 1250x^3 + ...

How do you find the coefficients of the power series for f(x) = 10/(1-5x)?

To find the coefficients of the power series for f(x) = 10/(1-5x), you can use the formula c_n = a^n * b^n, where a and b are the coefficients of the original function and n is the power of x.

What is the radius of convergence for the power series of f(x) = 10/(1-5x)?

The radius of convergence for the power series of f(x) = 10/(1-5x) is 1/5, since it is centered at x = 0 and the power series converges when |x| < 1/5.

How do you determine the interval of convergence for the power series of f(x) = 10/(1-5x)?

You can use the ratio test to determine the interval of convergence for the power series of f(x) = 10/(1-5x). The series will converge when the limit of |c_n+1/c_n| as n approaches infinity is less than 1. This will give you the interval of convergence centered at x = 0.

What is the function represented by the power series of f(x) = 10/(1-5x)?

The function represented by the power series of f(x) = 10/(1-5x) is the original function itself, since the power series is derived from the function using the formula for geometric series. Therefore, the power series is equal to the function for all values of x within the interval of convergence.

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