Question

Prove this theorem.

Answer 1

Answer:

Proved!

Step-by-step explanation:

$F(s)=\int\limits^\infty_0 f(t)e^{-st}dt=\\\\=\sum\limits^\infty_{n=0}\int\limits^{(n+1)p}_{np} f(t)e^{-st}dt=\\\\=\sum\limits^\infty_{n=0}\int\limits^p_0 f(u+np)e^{-su-snp}du= \: \: \: ,(u=t-np)\\\\=\sum\limits^\infty_{n=0} e^{-snp} \int\limits^p_0f(u)e^{-su}du=\\\\=(\int\limits^p_0f(u)e^{-su}du)\sum\limits^\infty_{n=0} e^{-snp}=\\\\=\frac{\int\limits^p_0f(u)e^{-su}du}{1-e^{-ps}}$