Reciprocal over a summation Is this statement true? Can we take reciprocal over a summation? 1 ∑ n = 1 ∞ 1 ( n + 1 ) 3 = ∑ n = 1 ∞ ( n + 1 ) 3

Question

Reciprocal over a summation Is this statement true? Can we take reciprocal over a summation?      1                  ∑                  n          =          1                ∞                    1                  (          n          +          1                      )            3                                =      ∑          n      =      1        ∞    (  n  +  1      )    3

talhekh · Accepted Answer

It is not true. By simple inspection, the sum in the denominator on the left side is a finite number, so the left side is finite. But the right side is apparently infinite.

lilmoore11p8 · Accepted Answer

No.       ∑          n      =      1        ∞        1          (      n      +      1              )        3              =  ζ  (  3  )  −  1  so it&#039;s reciprocal is       1          ζ      (      3      )      −      1       which is a finite number. The term on the right is obviously divergent.For reference,   ζ  (  3  )  ≈  1.20205690315959... and is called Apery&#039;s constant.

Reciprocal over a summation Is this statement true? Can we take reciprocal over a summation?

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