Studying the convergence of a series

I would love to research the merging of the collection: $$ \sum u_{n}$$ where $$ u_{n}=a^{s_{n}}$$

$$ s_{n}=\sum_{k=1}^n \frac{1}{k^b}$$ $$ a,b<1$$

We have:

$$ u_{n}=\exp((\frac{n^{1-b}}{1-b}+o(n^{1-b}))\ln(a))=\exp(\frac{\ln(a)n^{1-b}}{1-b}+o(n^{1-b}))$$

However $$\exp(o(n^{1-b}))$$ has to be defined. So just how can I establish: $$ o(n^{1-b})=s_{n}-\frac{n^{1-b}}{1-b}=\sum_{k=1}^n \frac{1}{k^b}-\frac{n^{1-b}}{1-b}$$?

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2022-07-25 20:40:31
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