Finding the Function Represented by a Power Series

[popo902]

Homework Statement

Determine the a_n 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

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

so...is this right?
if it is, where do i go from here?

 

[Dick]

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 />

 

[Sethric]

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.

 

[popo902]

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?