🏗️ $\Theta \rho ϵ\eta \Pi \alpha \tau \pi$🚧 (under construction)

Let $\{a_n\}$ be a sequence of complex numbers and $f(t)$ be a continuously differentiable function. Let $A(t)={\sum }_{n\le t}{a}_n$, then ${\sum }_{n\le x}{a}_{n}f(n)=A(x)f(x)-{\int }_1^{x}A(t)f^{\prime }(x)dt$

Let $0\le {\lambda }_1<{\lambda }_2<...$ be an increasing real sequence and $C_n$ be a complex sequence. Let $C(t)={\sum }_{{\lambda }_n\le t}{C}_n$ and $f(t)$ be a continuous derivative, then for $x\ge {\lambda }_1$, ${\sum }_{{\lambda }_n\le x}{C}_{n}f({\lambda }_n)=C(x)f(x)-{\int }_{{\lambda }_1}^{x}C(t)f^{\prime }(x)dt$ and if $C(x)f(x)\to 0$ as $x\to \infty$, then ${\sum }_{n=1}^{\infty }Cnf({\lambda }_n)=-{\int }_{\infty }^{{\lambda }_1}C(t)f^{\prime }(t)dt$