Given : S n = ∑ k = 1 n 1 k , T n = ∫ 1 n + 1 1 x d xFind a value of n so that S n ≥ 10

Question

Given :      S          n        =      ∑          k      =      1        n        1    k     ,       T          n        =      ∫          1              n      +      1            1    x    d  xFind a value of n so that       S          n        ≥  10

lodosr · Accepted Answer

By condensation, I think the solution means the idea behind Cauchy&#039;s condensation test. Let       S    n    =            ∑              k        =        1            n        f    (    k    )  , where   f  (  k  )  =            1      k      . We notice f is decreasing. Take a look in       S    n   when n is a power of 2      S    1    =  f  (  1  )      S    2    =  f  (  1  )  +  f  (  2  )      S    4    =  f  (  1  )  +  f  (  2  )  +  f  (  3  )  +  f  (  4  )  &amp;gt;  f  (  1  )  +  f  (  2  )  +  f  (  4  )  +  f  (  4  )Here we changed   f  (  3  ) for   f  (  4  ), since   f  (  3  )  &amp;gt;  f  (  4  )      S    8    =  f  (  1  )  +  f  (  2  )  +  ⋯  +  f  (  8  )  &amp;gt;  f  (  1  )  +  f  (  2  )  +  2  f  (  4  )  +  4  f  (  8  )Here we changed   f  (  3  ) for   f  (  4  ) and   f  (  5  )  ,  f  (  6  )  ,  f  (  7  ) for   f  (  8  ). And so on. More generally we have:      S                  2        N              &amp;gt;  f  (  1  )  +  f  (  2  )  +  2  f  (  4  )  +  4  f  (  8  )  +  8  f  (  16  )  +  ⋯  +      2          N      −      1        f  (      2    N    )  =  1  +            1      2        +            2      4        +            4      8        +            8      16        +  ⋯  +                    2                  N          −          1                            2        N              =  1  +                                                        1              2                                +                                    1              2                                +          ⋯          +                                    1              2                                      ⏟                    N        =  1  +            N      2      Then       S                  2        N              &amp;gt;  1  +            N      2      . Using   N  =  18 we get       S                  2                  18                      &amp;gt;  1  +            18      2        =  10

Given : S <mrow class="MJX-TeXAtom-ORD"> n </mrow> </msub> = <munderove

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