Representation of function as power series

aero_zeppelin
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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?
 
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aero_zeppelin said:

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

Any help please?
 
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
 
aero_zeppelin said:
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.
 
Got it... Thanks so much!
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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