∑ k = 2 ∞ ( 1 − e − n s k − 1 ) k Compute an approximation of this series, for the regime where n is large.

Question

∑                      k            =            2                    ∞                                      (            1            −                          e                              −                n                                  s                                      k                    −                    1                                                                        )                    k                    Compute an approximation of this series, for the regime where n is large.

taweirrvb · Accepted Answer

Keep the exponential form and do binomial expansion for say   k  ≤  4 (you may want to try a different number) and ignore terms for   5  &amp;lt;  k. Then the form would be                              ∑                      k            =            2                                ∞                                                (            1            −                          e                              −                n                                  s                                      k                    −                    1                                                                        )                    k                                    ≈        1        −                  e                      −            n            s                          +        1        −        2                  e                      −            n                          s              2                                      +                  e                      −            2            n                          s              2                                      +        1        −        3                  e                      −            n                          s              3                                      +        3                  e                      −            2            n                          s              3                                      −                  e                      −            3            n                          s              3

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