Real series where sum equals -1 and weighted sum equals 0

[dimitri151]

Find a series of real numbers {a_n} such that a₁+ a₂+a₃+...=-1 and a₁+3 a₂+5 a₃+...+(2n-1)a_n+...=0.
Also there have to be an infinite number of non-zero a_ns.

I'm not sure how to begin this. Can it be done?

 

[I like Serena]

Find a series of real numbers {a_n} such that a₁+ a₂+a₃+...=-1 and a₁+3 a₂+5 a₃+...+(2n-1)a_n+...=0.
Also there have to be an infinite number of non-zero a_ns.

I'm not sure how to begin this. Can it be done?

Suppose you subtract the first series from the second.
See any pattern?