Finding the Function Represented by a Power Series

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Homework Statement



Determine the an so that the equation
\sum_{n=1}^{\infty}{na_{n}x^{n-1}} + 2\sum_{n=0}^{\infty}{a_{n}x^{n}} = 0<br />


is satisfied. Try to identify the function represented by the series
\sum_{n=0}^{\infty}{a_{n}x^{n}} = 0<br />


Homework Equations





The Attempt at a Solution




what i have so far is

\sum_{n=0}^{\infty}x^{n}[{a_{n+1}(n+1) + 2a_{n}}]= 0<br />

i just combined the series.
then i solved for a,n

an = -1/2(an+1)(n+1)

so...is this right?
if it is, where do i go from here?
 
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I would write it as a_{n+1}=(-2/(n+1))*a_{n}. But sure, that's ok. To identify the function I'd notice that one of those sums looks like the derivative of the other sum. Try and write a differential equation for
<br /> f(x)=\sum_{n=0}^{\infty}{a_{n}x^{n}}<br />
 
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Or alternatively, you can use the original given equation to find a_0, then derive a formula for a_n from that. It sounds like that's what they want. The differential equation method mentioned by Dick, however, is much quicker if you are allowed that.
 
i noticed from the first equation that the left sum, was the deriv. of the right, ecxept that the 2 was in front
so technically it would look like this : y' + 2y = 0, when y = \sum_{n=0}^{\infty}{a_{n}x^{n}} = 0<br />

so are you saying i should just solve the normal DE and i'll get the solution that resembles that summation...?


"I would write it as a_{n+1}=(-2/(n+1))*a_{n}."
i see now that this way was better since you need a0 to find the rest
...but then i thought i was solving for an?
 
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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