Prove that

$\displaystyle \displaystyle\lim_{x\to\infty} \frac{2}{(\ln(x))^2} \sum\limits_{1 \leq k \leq x-1} \frac{\ln k}{k+1} = 1$

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- Oct 26th 2010, 01:22 AMNewtonianProve the limit of this sequence is 1
Prove that

$\displaystyle \displaystyle\lim_{x\to\infty} \frac{2}{(\ln(x))^2} \sum\limits_{1 \leq k \leq x-1} \frac{\ln k}{k+1} = 1$ - Oct 26th 2010, 11:39 AMzzzoak
As x goes to infinity the sum

$\displaystyle

\displaystyle { \sum\limits_{1 \leq k \leq x-1} \frac{\ln k}{k+1} }

$

may be evaluated

$\displaystyle

\displaystyle { \int_{1}^{x} \frac{ln \; t}{t+1}dt= \int_{1}^{x} ln \; t \:\; d( \; ln(t+1) \;) \; =ln \; t \:\; ln(t+1) \; |_1^x - \int_{1}^{x} \frac{ln \; (t+1)}{t}dt

}

$

$\displaystyle

2 \; \displaystyle { \int_{1}^{x} \frac{ln \; t}{t+1}dt= ln^2x

$ - Oct 26th 2010, 01:26 PMNewtonian
Sorry, I don't follow what you're trying to say. The sum may be approximated by the integral, but they're not equal, and the integral isn't equal to $\displaystyle (\ln(x))^2$ so I'm not quite sure what you're trying to show...could you explain a bit more please?

- Oct 26th 2010, 03:21 PMzzzoak
This sum is a Riemann sum of this integral.

For x going to infinity

$\displaystyle

ln(1+x)=ln(x(1+1/x))=lnx+ln(1+1/x)=lnx+1/x +o(1/x)

$ - Oct 30th 2010, 08:19 AMNewtonian
OK I see what you're saying now. I think everything's fine as long as we can show that the approximation of the sum by the integral at the beginning is valid; that is, we need to show

$\displaystyle \sum\limits_{1 \leq k \leq x-1} \frac{\ln k}{k+1} \sim \int_{1}^{x} \frac{\ln(t)}{t+1}$

as $\displaystyle x \rightarrow \infty$, where $\displaystyle f(x) \sim g(x)$ means $\displaystyle f(x)/g(x) \rightarrow 1$ as $\displaystyle x \rightarrow \infty$. I've thought about this for a while; is it just a standard kind of integration result or can you prove it?