Let $\lambda_n$ be a sequence of positive numbers and $a_n$ be a sequence of real numbers such that
$$\sum \limits_{i=1}^{\infty} a_n \lambda_n^{k}=0 \ \ \hbox{for all}\ \ k\geq 0.$$
Is $a_n=0$ for all $n$?
An elementary question about a sequence of numbers
A random mathematician
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