Question Number 148609 by learner001 last updated on 29/Jul/21

prove that (a_n )_(n≥1 ) defined by a_n =(1/2)+(1/6)+...+(1/(n(n+1))) is cauchy sequence.

Commented bylearner001 last updated on 29/Jul/21

This is what i tried. let ε>0 be given i need an n^∗ such that p∈N if ∀ n≥n^∗ then ∣a_(n+p) −a_n ∣<ε. ∣a_(n+p) −a_n ∣=∣((1/(n+1))−(1/(n+2)))+((1/(n+2))−(1/(n+3)))+...+((1/(n+p))−(1/(n+p+1)))∣ =∣(1/(n+1))−(1/(n+p+1))∣≤∣(1/(n+1))∣+∣(1/(n+p+1))∣<(1/n)+(1/(n+p))<(1/n)<ε if n^∗ ≥(1/ε) then ∣a_(n+p) −a_n ∣<ε ∀ n≥n^∗ .

Commented bylearner001 last updated on 29/Jul/21

is this correct?