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let-u-n-1-1-2-1-3-1-n-prove-that-u-n-is-divdrgente-




Question Number 57412 by Abdo msup. last updated on 03/Apr/19
let u_n =1 +(1/( (√2))) +(1/( (√3))) +...+(1/( (√n)))  prove that (u_n ) is divdrgente.
letun=1+12+13++1nprovethat(un)isdivdrgente.
Commented by maxmathsup by imad last updated on 07/Apr/19
let f(x)=(√x)   ∃c ∈]k,k+1[   /f(k+1)−f(k)=(k+1−k)f^′ (c)⇒(√(k+1)) −(√k)=(1/(2(√c)))  we have    k<c<k+1 ⇒(√k)<(√c)<(√(k+1)) ⇒(1/(2(√c))) <(1/(2(√k))) ⇒  (1/(2(√k))) >(√(k+1))−(√k) ⇒ Σ_(k=1) ^n  (1/( (√k))) >2 Σ_(k=1) ^n  {(√(k+1))−(√k)) ⇒  U_(n )  > 2{(√(n+1))−1} →+∞ (n→+∞) ⇒U_n  diverges and lim_(n→+∞) U_n =+∞ .
letf(x)=xc]k,k+1[/f(k+1)f(k)=(k+1k)f(c)k+1k=12cwehavek<c<k+1k<c<k+112c<12k12k>k+1kk=1n1k>2k=1n{k+1k)Un>2{n+11}+(n+)Undivergesandlimn+Un=+.

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