Power Series Representation Problem

Jan 2010
13
1
Okay so if x - ((x^3)/3!) + ((x^5)/5!) - ((x^7)/7!) + ... + (((-1)^n)(x^(2n+1)))/(2n+1)! represents sinx, what is the series representation for (x^3)sin(2x)?
 
Oct 2009
4,261
1,836
Okay so if x - ((x^3)/3!) + ((x^5)/5!) - ((x^7)/7!) + ... + (((-1)^n)(x^(2n+1)))/(2n+1)! represents sinx, what is the series representation for (x^3)sin(2x)?

\(\displaystyle x^3\sin(2x)=x^3\sum^\infty_{n=1}(-1)^{n-1}\frac{(2x)^{2n-1}}{(2n-1)!}=\sum^\infty_{n=1}(-1)^{n-1}\frac{2^{2n-1}x^{2n+2}}{(2n-1)!}\)

Tonio
 
Jan 2010
13
1
Hmm. I'm still confused. Where did the 2^(2n-1) come from? I just don't understand what to change about the given series to make it represent the new series. Sorry to be difficult, this is just a tough topic for me.
 
Oct 2009
4,261
1,836
Hmm. I'm still confused. Where did the 2^(2n-1) come from? I just don't understand what to change about the given series to make it represent the new series. Sorry to be difficult, this is just a tough topic for me.

\(\displaystyle (ab)^n=a^nb^n\Longrightarrow (2x)^{2n-1}=2^{2n-1}x^{2n-1}\) ...simple HS (powers) algebra that we must remember and master.

Tonio