Determine Elementary Fcn for sum[1,infty](n a_n x^{n-1}) - sum[1,infty](a_n x^n) = 0

[jules1234]

3. Determine an expression for \(\displaystyle a_n\) so that the following is satisfied:

. . . . .\(\displaystyle \displaystyle \sum_{n=1}^{\infty}\, \left(n\, a_n\, x^{n-1}\right)\, -\, 3\, \sum_{n=0}^{\infty}\, \left(a_n\, x^n\right)\, =\, 0\)

and identify the elementary function defined by the power series

. . . . .\(\displaystyle \displaystyle y(x)\, =\, \sum_{n=0}^{\infty}\, a_n\, x^n\)

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I have figured out the expression for an should be:

a_n=(3^(n+1) a_0)/n!Xn

I also realize that e^x= the power sum from 0 to infinity of X^n/n!

I am not sure how to write this a_n as the elementary function e^x

 

[Deleted member 4993]

3. Determine an expression for \(\displaystyle a_n\) so that the following is satisfied:

. . . . .\(\displaystyle \displaystyle \sum_{n=1}^{\infty}\, \left(n\, a_n\, x^{n-1}\right)\, -\, 3\, \sum_{n=0}^{\infty}\, \left(a_n\, x^n\right)\, =\, 0\)

and identify the elementary function defined by the power series

. . . . .\(\displaystyle \displaystyle y(x)\, =\, \sum_{n=0}^{\infty}\, a_n\, x^n\)

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I have figured out the expression for an should be:

a_n=(3^(n+1) a_0)/n!Xn

I also realize that e^x= the power sum from 0 to infinity of X^n/n!

I am not sure how to write this a_n as the elementary function e^x

If I were to do this problem, I would write out the first few terms (~4) for each series and observe if any grouping or elimination is possible.

 

[HallsofIvy]

Another way to do this is to note that \(\displaystyle \sum_{n=1}^\infty na_nx^{n-1}\) is the derivative of \(\displaystyle \sum_{n= 0}^\infty a^nx^n\) so letting \(\displaystyle f(x)= \sum_{n= 0}^\infty (a^n * x^n)\) that equation becomes \(\displaystyle f'- 3f= 0\). What functions satisfy that differential equation?