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lim-n-1-n-1-2-3-3-n-n-




Question Number 174951 by infinityaction last updated on 14/Aug/22
  lim_(n→∞) (1/n)[1+(√2)+^3 (√3)+...^n (√n)]
limn1n[1+2+33+nn]
Answered by TheHoneyCat last updated on 16/Aug/22
There′s a theorem that sates that  U_n   →_(n→∞) l ∈ R^_  ⇒ lim_(n→∞) Σ_(k=0) ^n U_k =l    Now  ^n (√n)=n^(1/n) =exp(((ln n)/n))   since ln n /n→0  ^n (√n)→exp(0)=1      Thus:   your limit is 1.
TheresatheoremthatsatesthatUnnlR_limnnk=0Uk=lNownn=n1/n=exp(lnnn)sincelnn/n0nnexp(0)=1Thus:yourlimitis1.
Commented by TheHoneyCat last updated on 16/Aug/22
Woops, didn′t pay attention  the theorem states that   lim_(n→∞) (1/n)Σ_(k=0) ^n U_k =l  of course... Otherwise it wouldn′t work...  Sorry
Woops,didntpayattentionthetheoremstatesthatlimn1nnk=0Uk=lofcourseOtherwiseitwouldntworkSorry

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