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Hi everybody,

while doing a complex analysis exercise, i came to a strange inequality which i don't know how to interpretate. Suppose you have a sequence $\{a_j\}$ of positive real number. Let $\rho$ a positive real number. The inequality i found after some calculation is

$$\sum_{j=1}^{+\infty}\frac{1}{|a_j|^{\rho +\epsilon}}\leq \sum_{j=1}^{+\infty}\frac{1}{|a_j|^{\rho-\epsilon}}$$

for every $\epsilon>0$.

My question is: can i deduce something from this inequality? for example the convergence of the first series (that with $+\epsilon$)? Can i deduce nothing? Is that inequality surely false?

Kind regards

while doing a complex analysis exercise, i came to a strange inequality which i don't know how to interpretate. Suppose you have a sequence $\{a_j\}$ of positive real number. Let $\rho$ a positive real number. The inequality i found after some calculation is

$$\sum_{j=1}^{+\infty}\frac{1}{|a_j|^{\rho +\epsilon}}\leq \sum_{j=1}^{+\infty}\frac{1}{|a_j|^{\rho-\epsilon}}$$

for every $\epsilon>0$.

My question is: can i deduce something from this inequality? for example the convergence of the first series (that with $+\epsilon$)? Can i deduce nothing? Is that inequality surely false?

**EDIT:**the sequence $a_j$ tends to $\infty$Kind regards

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