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Prove or disprove: If the sequence $(x_{n})_{n\in\mathbb{N}}\subset...
Prove or disprove: If the sequence $(x_{n})_{n\in\mathbb{N}}\subset\mathbb{R}$ is convergent then $(nx_{n})_{n\in\mathbb{N}}$ is divergent.The statement is true.(It would work for some exceptions, like...
View ArticleAnswer by Zau for Prove or disprove: If the sequence...
The statement is not true. If $x_n = 0$ which is convergent then $n x_n = 0$ which is also convergent.For the non-zero case it is true. If $\lim_{n\to +\infty}x_n =c$, then $\lim_{n\to +\infty}n x_n...
View ArticleAnswer by Andres Mejia for Prove or disprove: If the sequence...
Suppose wlog that as $n \to \infty $, $x_n \to L > 0$. Then there exists some $N$ so that $n \geq N $ implies that $x_n> L/2$.Then $n x_n > n (\frac {L}{2}) $, which diverges.
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