Return to Question

Let $\lambda_n$ be an increasing and unbounded sequence of positive real numbers and $a_n$ be a sequence of real numbers such that

$$\sum \limits_{n=1}^{\infty} a_n \lambda_n^{k}=0 \ \ \hbox{for all}\ \ k\geq 0.$$

Is $a_n=0$ for all $n$?

Let $\lambda_n$ be an increasing and unbounded sequence of positive real numbers and $a_n$ be a sequence of real numbers such that

$$\sum_{n=1}^\infty a_n \lambda_n^k=0 \ \ \text{ for all }\ \ k\geq 0.$$

Is $a_n=0$ for all $n$?

Let $\lambda_n$ be an increasing and unbounded sequence of positive numbers and $a_n$ be a sequence of real numbers such that

$$\sum \limits_{n=1}^{\infty} a_n \lambda_n^{k}=0 \ \ \hbox{for all}\ \ k\geq 0.$$

Is $a_n=0$ for all $n$?

Let $\lambda_n$ be an increasing and unbounded sequence of positive real numbers and $a_n$ be a sequence of real numbers such that

$$\sum \limits_{n=1}^{\infty} a_n \lambda_n^{k}=0 \ \ \hbox{for all}\ \ k\geq 0.$$

Is $a_n=0$ for all $n$?

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 $n$?

Let $\lambda_n$ be an increasing and unbounded sequence of positive numbers and $a_n$ be a sequence of real numbers such that

$$\sum \limits_{n=1}^{\infty} a_n \lambda_n^{k}=0 \ \ \hbox{for all}\ \ k\geq 0.$$

Is $a_n=0$ for all $n$?