Asymptotics of the solution of a differential equation

Let f be twice continuously differentiable function defined on \mathbb{R}, such that f(x)f''(x)=1 for all x\ge 0, f(0)=1 and f'(0)=0.

Find \lim_{x\to\infty} \frac{f(x)}{x\sqrt{\ln(x)}}.


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Basel problem type sum, proposed by prof. Skordev

Evaluate \sum_{n=1}^{\infty}\frac{1}{2^n n^2}.
Proof. The function S(x):=\sum_{n=1}^{\infty}\frac{x^n}{ n^2}
is well defined for x\in[-1,1] and thus we need to find S(1/2). Using differentiation, we observe that S(x)=-\int_0^x\frac{\log(1-t)}{t}\text{d}t.
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On the other hand, using integration by parts, we obtain S(x)=-\int_0^x\log(1-t)\text{d}\log t=-\log t\log(1-t)\Big|_{t=0}^x-\int_0^x\frac{\log t}{1-t}\text{d}t.
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A problem proposed by prof. Babev

Consider the sequence a_n=\int_{n}^{n+1}\frac{\sin^2(\pi t)}{t}\text{d}t.
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Modification on a sequence from VJIMC

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