The sum of series ( log ⁡ 3 ) 1 1 ! + ( log ⁡ 3 ) 3 3 ! + ( log ⁡ 3 ) 5 5 ! + ⋯ is what? Is there a general algorithm to find the summation of logarithms?

Question

The sum of series             (      log      ⁡      3              )        1                    1      !        +            (      log      ⁡      3              )        3                    3      !        +            (      log      ⁡      3              )        5                    5      !        +  ⋯ is what? Is there a general algorithm to find the summation of logarithms?

Brogan Navarro · Accepted Answer

The series itself does not have too much to do with logarithms; to see why without getting lost with the log everywhere, let   α  =  log  ⁡  3. You want      ∑          n      =      0        ∞              (      log      ⁡      3              )                  2          n          +          1                            (      2      n      +      1      )      !        =      ∑          n      =      0        ∞              α              2        n        +        1                    (      2      n      +      1      )      !        =  sinh  ⁡  αby the series definition of   sinh. That being said, now here we have simplifications because   α  =  log  ⁡  3. Indeed, recall that, for every   x  ∈      R  ,  sinh  ⁡  x  =                    e        x            −              e                  −          x                      2  and therefore here  sinh  ⁡  log  ⁡  3  =                    e                  log          ⁡          3                    −              e                  −          log          ⁡          3                      2    =            3      −      1              /            3        2    =                                          4            3                                  .

The sum of series ((log 3)^1)/(1!)+((log 3)^3)/(3!)+((log 3)^5)/(5!)+... is what? Is there a general algorithm to find the summation of logarithms?

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