I was solving a problem and I had to compute ∑ n = 1 ∞ n ⋅ ( 1.05 1.1 ) n .

Question

I was solving a problem and I had to compute      ∑          n      =      1              ∞        n  ⋅            (              1.05        1.1            )        n    .

Johnathan Morse · Accepted Answer

With calculus. Let   f  (  x  )  =      ∑          n      ≥      1            x    n  ; this is the sum of an infinite geometric series with ratio x and first term x, so we know that                    (1)                    f        (        x        )        =                  ∑                      n            ≥            1                                    x          n                =                  x                      1            −            x                                  ,            provided that       |    x      |    &amp;lt;  1. Now differentiate (1):  f          ′    (  x  )  =      ∑          n      ≥      1        n      x          n      −      1        =      1          (      1      −      x              )        2                .That sum is almost what you want, and if we multiply by x, we’ll have what you want:                    (3)                              ∑                      n            ≥            1                          n                  x          n                =        x                  ∑                      n            ≥            1                          n                  x                      n            −            1                          =                  x                      (            1            −            x                          )              2                                              .            Substitute   x  =      1.05    1.1   into (3), and you’ll get the desired value.

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