What is the sum of the first $n$-terms of the series, whose $k$-term is $k ! imes k$ ?

Question

What is the sum of the first $n$-terms of the series, whose $k$-term is $k ! 	imes k$ ?

Aicharya · Accepted Answer

Given, $\begin{aligned} t_k & =k k !=(k+1-1) k ! \ & =(k+1) k !-k !=(k+1) !-k !\end{aligned}$ $\begin{aligned} & t_1=2 !-1 ! \ & t_2=3 !-2 ! \ & t_3=4 !-3 !\end{aligned}$ ......... ......... $\underline{t_n=(n+1) !-n !}$ $t_1+t_2+\ldots .+t_n=(n+1) !-1$ $	herefore$ Required sum $=(n+1) !-1$

What is the sum of the first $n$-terms of the series, whose $k$-term is $k ! \times k$ ?

Solution