Question-81768

Question Number 81768 by Power last updated on 15/Feb/20

Commented by mathmax by abdo last updated on 15/Feb/20

S_n =Σ_(k=1) ^n (k/((k+1)!)) =Σ_(k=1) ^n ((k+1−1)/((k+1)!)) =Σ_(k=1) ^n ((1/(k!))−(1/((k+1)!))) =1−(1/(2!)) +(1/(2!))−(1/(3!)) +...+(1/(n!))−(1/((n+1)!)) =1−(1/((n+1)!))

$${S}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{{k}}{\left({k}+\mathrm{1}\right)!}\:=\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{{k}+\mathrm{1}−\mathrm{1}}{\left({k}+\mathrm{1}\right)!}\:=\sum_{{k}=\mathrm{1}} ^{{n}} \left(\frac{\mathrm{1}}{{k}!}−\frac{\mathrm{1}}{\left({k}+\mathrm{1}\right)!}\right) \\ $$$$=\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}!}\:+\frac{\mathrm{1}}{\mathrm{2}!}−\frac{\mathrm{1}}{\mathrm{3}!}\:+…+\frac{\mathrm{1}}{{n}!}−\frac{\mathrm{1}}{\left({n}+\mathrm{1}\right)!}\:=\mathrm{1}−\frac{\mathrm{1}}{\left({n}+\mathrm{1}\right)!} \\ $$

Commented by mr W last updated on 15/Feb/20