Find Real Series: a1+ a2+a3+...=-1 & a1+3 a2+5 a3+...+(2n-1)an+...=0

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The discussion focuses on finding a series of real numbers {an} that satisfies two conditions: the sum a1 + a2 + a3 + ... = -1 and the weighted sum a1 + 3a2 + 5a3 + ... + (2n-1)an + ... = 0. Participants suggest exploring the relationship between these two series by subtracting the first from the second to identify potential patterns. The requirement for an infinite number of non-zero terms in the series adds complexity to the problem, indicating a need for advanced mathematical techniques.

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

I'm not sure how to begin this. Can it be done?
 
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dimitri151 said:
Find a series of real numbers {an} such that a1+ a2+a3+...=-1 and a1+3 a2+5 a3+...+(2n-1)an+...=0.
Also there have to be an infinite number of non-zero ans.

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

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

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