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

$\displaystyle

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

$

and deduce that $\displaystyle \sum_{n \in Z} f(x+n) $ converges

prove that

https://nrich.maths.org/tex2gif/tex2...B-x%5E2%2F2%7D

this is based on the assumption that https://nrich.maths.org/tex2gif/tex2...%5E2%7D%7B2%7D 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

https://nrich.maths.org/tex2gif/tex2...%5E-x%20d%5Cmu (i have done this part)

deduce that

https://nrich.maths.org/tex2gif/tex2...%2B1%2Fn%29%29

any thoughts would be appreciated! thank you!