Is it possible to find complex numbers Cn, so that both equations are satisfied?

[LagrangeEuler]

Is it possible two find complex numbers $C_n$, so that both equations are satisfied
$$\sum^{\infty}_{n=1}nC_n=0$$
and
$$\sum^{\infty}_{n=1}|C_n|^2=1$$?

 

[BvU]

$C_1=2/\sqrt 5,\quad C_2 = -1/\sqrt 5,\quad C_{n>2}=0$

 

[LagrangeEuler]

Thanks a lot. One more question. Is it possible to find constants that satisfy three conditions
$$\sum^{\infty}_{n=1}|C_n|^2=1$$
$$\sum^{\infty}_{n=1}nC_n=0$$
$$\sum^{\infty}_{n=1}(-1)^nnC_n=0$$

 

[BvU]

Are we doing your homework for you ? What is the context of these exercises ?

 

[LagrangeEuler]

No. I am trying to find wave function of infinite potential well which derivative does not have jump in the boundaries. To my mind this is only possible if we constants that satisfy those three equation. I do not have idea how to find them.

 

[BvU]

With an infinite well there is a jump in the derivative

 

[greswd]

With an infinite well there is a jump in the derivative

and for finite wells, the derivative must not jump right?

 

[Infrared]

How about $C_1=3/\sqrt{10}, C_2=0, C_3=-1/\sqrt{10}$ and $C_n=0$ for $n>3$?