Question Number 103673 by bobhans last updated on 16/Jul/20

Σ_(k=1) ^(4095) (1/(((√k)+(√(k+1)))((k)^(1/4) +((k+1))^(1/4) ))) ?

Answered by bramlex last updated on 16/Jul/20

(1/(((√k)+(√(k+1)))((k)^(1/4) +((k+1))^(1/4) ))) ×((((k+1))^(1/4) −(k)^(1/4) )/(((k+1))^(1/4) −(k)^(1/4) ))=  ((((k+1))^(1/4) −(k)^(1/4) )/(((√(k+1))+(√k))((√(k+1))−(√k)))) = ((k+1))^(1/4) −(k)^(1/4)   now S= Σ_(k=1) ^(4095) (((k+1))^(1/4) −(k)^(1/4) ) =((4096))^(1/4) −1=7  (note telecoping series)