Show that for every integer 1 + 1 4 + 1 9 + · · · + 1 n 2 ≤ 2 − 1 n

Question

Show that for every integer    1  +      1    4    +      1    9    +  ·  ·  ·  +      1          n      2        ≤  2  −      1    n

Kiana Cantu · Accepted Answer

Hint: Note that      1          2      2        +      1          3      2        +  ⋯  +      1          n      2        &amp;lt;      1          1      ⋅      2        +      1          2      ⋅      3        +  ⋯  +      1          (      n      −      1      )      ⋅      n        .The expression on the right turns out to be a telescoping sum. For       1          (      i      −      1      )      ⋅      i        =      1          i      −      1        −      1    i  One could also do an induction version of the above proof, using the fact that       1          (      n      +      1              )        2              &amp;lt;      1    n    −      1          n      +      1

ban1ka1u · Accepted Answer

Consider a Riemann sum approximation for      ∫    1    n        1          x      2        d  xby means of the partition   {  1  ,  2  ,  3  ,  …  ,  n  } and a right endpoint approximation. Since       1          x      2       is decreasing, the approximation will give an underestimate for the integral.We obtain      1    4    +      1    9    +  ⋯  +      1          n      2        ≤      ∫    1    n        1          x      2        d  x  =            [      −              1        x            ]        1    n    =  1  −      1    n  The result in the problem follows immediately.