Representation of function as power series

[aero_zeppelin]

Homework Statement

Represent 5x / (10 + x) as a power series, find c0, c1, c2, c3 and c4, find the radius of convergence

The Attempt at a Solution

I think I got the representation fine:

Ʃ from 0 to ∞ = 1/2 [ (-1^n)(x^n+1) / 10^n ]

Radius of Conv. = 10

But what the hell are those c0, c1 etc...? I thought they were the coefficients of the series but I keep getting them wrong...

My answers are: c0 = 1/2 , c1 = -1/20, c2 = 1/200 ...

Any help please?

 

[Mark44]

Homework Statement

Represent 5x / (10 + x) as a power series, find c0, c1, c2, c3 and c4, find the radius of convergence

The Attempt at a Solution

I think I got the representation fine:

Ʃ from 0 to ∞ = 1/2 [ (-1^n)(x^n+1) / 10^n ]

Radius of Conv. = 10

But what the hell are those c0, c1 etc...? I thought they were the coefficients of the series but I keep getting them wrong...

They are the coefficients of the series, with c₀ being the constant term, c₁ the coefficient of x, c₂ the coefficient of x², and so forth.

It looks like your coefficients are out of sync. You have what appear to be the right numbers, but the subscripts are off by one.

My answers are: c0 = 1/2 , c1 = -1/20, c2 = 1/200 ...

Any help please?

 

[aero_zeppelin]

It's been driving me crazy haha. Ok, so it should be:

c0 = ?
c1 = 1/2
c2 = -1/20
c3 = 1/200
c4 = -1/2000

So you just plug in "n" and see the respective coefficients? What happens with c0?

Thanks a lot

 

[Mark44]

It's been driving me crazy haha. Ok, so it should be:

c0 = ?
c1 = 1/2
c2 = -1/20
c3 = 1/200
c4 = -1/2000

So you just plug in "n" and see the respective coefficients? What happens with c0?

Thanks a lot

c₀ = 0

I didn't verify that your formula is correct, as I did this by dividing 5x by 10 + x using polynomial long division. The first few terms I got were (1/2)x - (1/20)x² + (1/200)x³, and these coefficients agree with the ones you show.

Since there is no constant term, that means its coefficient is zero.

 

[aero_zeppelin]

Got it... Thanks so much!