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alexthepenguin · #1 November 25th, 2008, 02:23 AM

i have exams on this course coming up in two days (i should probably write this post earlier). i was hoping to read more and try to find a solution myself but unfortunately failed.
let f be a positive function, integrable over R. prove that

$\sum_{n \in Z} \int_0^1 f(x+n) d \mu = \int_{-\infty}^{\infty} f(x) d \mu$
and deduce that $\sum_{n \in Z} f(x+n)$ converges

prove that

$\lim_{n\rightarrow \infty} \sqrt{n} \int_0^{\pi/2} \cos^n x d\mu=\int_0^\infty e^{-x^2/2}$

this is based on the assumption that $\ln(\cos x) \le \frac{-x^2}{2}$ for all x in 0..pi/2 ( i have done this part)

also one mroe if p >-1 and q belongs to the natural number. prove that

$\lim_{n\rightarrow \infty} \int_0^n x^p(ln x)^q(1-x/n)^n d\mu=\lim_{n\rightarrow \infty} \int_0^\infty x^p (ln x)^q e^-x d\mu$ (i have done this part)

deduce that
$\int_0^\infty e^-x ln x d\mu =\lim_{n\rightarrow \infty}(ln x-(1+1/2+...+1/n))$

any thoughts would be appreciated! thank you!