Find Sum of Power Series: Hint and Tips

[PCSL]

I have to find the sum of the power series: $$\sum_{n=1}^\infty nx^{n+1}$$

I know the I'm supposed to show work but I don't have any idea where to start. I'm not asking for you to do the problem for me, just a hint.

The only idea I had was to take the derivative to get rid of the n+1 in the exponent but I'm not sure if $$\sum_{n=1}^\infty n(n+1)x^{n}$$ is any easier to solve.

Also I tried looking at the integral but again didn't see what to do
$$\sum_{n=1}^\infty \frac{nx^{n+2}}{n+2}$$

Thank you.

 

[awkward]

Try factoring out $x^2$ and see if the resulting series reminds you of something familiar.

 

[PCSL]

Try factoring out $x^2$ and see if the resulting series reminds you of something familiar.

So then I would have:

$$x^2\sum_{n=1}^\infty nx^{n-1}$$

That doesn't look familiar.. should it?

 

[SammyS]

So then I would have:

$$x^2\sum_{n=1}^\infty nx^{n-1}$$

That doesn't look familiar.. should it?

Integrate $\displaystyle nx^{n-1}$

 

[PCSL]

Integrate $\displaystyle nx^{n-1}$

I guess I just need more practice with these. Thank you, I don't think I would have thought of that any time soon ;).