We know that $$\sum_{k≥2} \frac{1}{k \log(k)} $$ diverges. How many terms must be taken before the partial sum exceed 10?
I have know idea which theorem or what kind of a way I should take this. Any hints?
Approximation of partial sums and series
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real-analysis
sequences-and-series
approximation
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0this does not make sense. if you meant what you wrote then the $k$s could cancel. can you check? – 2017-02-23
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0I have edited the formula for the sum. Is my edit correct? – 2017-02-23
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0Have you an idea what function is asymptotically equal to $$\sum_{2 \leqslant k \leqslant x} \frac{1}{k\log k}\,?$$ – 2017-02-23
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2Since $\int_{e}^{M}\frac{dx}{x\log x}=\log\log M$, we need about $\exp(\exp 10)$ (i.e. a colossal number) terms before the partial sums exceed $10$. – 2017-02-23
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0I found $e^{e^{10}}-1$ – 2017-02-23