Prove that if lim n → ∞ a n = 1 then lim n → ∞ a 1 + a 2 + … + a n = ∞

Question

Prove that if       lim          n      →      ∞            a          n        =  1 then       lim          n      →      ∞            a          1        +      a          2        +  …  +      a          n        =  ∞

grindweg1v · Accepted Answer

If       a    n    →  1 then there is   N  ∈      N   such that       a    n    ≥  1      /    2 for all   n  ≥  N. Therefore, for   n  ≥  N      b    n    =      ∑          k      =      1        n        a    k    ≥  (  n  −  N  )      1    2    +      ∑          k      =      1        N        a    k  and   (  n  −  N  )  1      /    2 goes to   ∞ for   n  →  ∞  .

Prove that if <munder> <mo movablelimits="true" form="prefix">lim <mrow class="MJX-TeXAto

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