(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Show that if [tex]f(x)[/tex] tends to zero monotonically as x increases without limit, and is continuous for x>0, and of the series [tex]\sum_{k=1}^{\infty}f(k)[/tex] diverges, then [tex]\sum_{k=1}^nf(k) \sim \int_1^nf(x)dx[/tex].

If g(x) is a second function satisfying the same hypotheses as f(x), and if g(x)=o(f(x)), show that

[tex]\sum_{k=1}^ng(k)=o(\sum_{k=1}^nf(k)_[/tex].

3. The attempt at a solution

I think by monotonicity we can conclude that

[tex]\frac{\int_1^nf(x+1)dx}{\sum_{k=1}^nf(k)}\le1\le\frac{f(1)+\int_1^nf(x)dx}{\sum_{k=1}^nf(k)}[/tex]

And I want to show that LHS and RHS converge to each other, hence they both converge to 1. But I can't get it to work.

The second part I don't know what to do.

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# Monotonous functions and asymptotes

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