Is there a relationship between ∑ n = 1 ∞ ( 1 2 ) n = 1 and ∫ 1 ∞ 1 x 2 d x = 1 ?

Question

Is there a relationship between       ∑          n      =      1              ∞                  (                        1          2                    )              n        =  1 and       ∫          1              ∞            1          x      2          d  x  =  1 ?

fugprurgeil · Accepted Answer

You can break the integral       ∫          1              ∞            1          x      2        d  x into a sum of integrals over intervals   [      2    n    ,      2          n      +      1        ]      ∫          1              ∞            1          x      2        d  x  =      ∑          n      =      0              ∞            ∫                  2        n                            2                  n          +          1                          1          x      2        d  x  =      ∑          n      =      0              ∞            (          1              2        n              −          1              2                  n          +          1                      )    =      ∑          n      =      0              ∞            1          2              n        +        1              =      ∑          n      =      1              ∞            1          2              n              .

Ellen Chang · Accepted Answer

Note that by the substitution   u  =  1      /    x, we have the integral      ∫    1    ∞        1          x      2        d  x  =      ∫    0    1    1  d  u  .Meanwhile, we have the series      ∑          n      =      1        ∞        1          2      n        =      ∑          n      =      0        ∞        1          2              n              −      1          2              n        +        1              .Both express the area of a unit block, the former by integrating, the latter by telescoping.

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New Questions in Calculus 1