Fix N ∈ N and define f ( N ) = ∑ n = N ∞ 1 2 n ( n N )

Question

Fix   N  ∈      N   and define  f  (  N  )  =      ∑          n      =      N        ∞        1          2      n                                    (                            n        N                              )

Bria Berg · Accepted Answer

We have  (  1  −  x      )          −      1        =      ∑          n      =      0        ∞        x    n  Thus  N  !    (  1  −  x      )          −      1      −      N        =                    d        N                    d                  x          N                      (  1  −  x      )          −      1        =                    N        !                    x        N                  ∑          n      =      N        ∞                                (                            n        N                              )                          x    n  Therefore      x    N    (  1  −  x      )          −      1      −      N        =      ∑          n      =      N        ∞                                (                            n        N                              )                          x    n        x    N    (  1  −  x      )          −      1      −      N        =      ∑          n      =      N        ∞                                (                            n        N                              )                          x    n  Plugging in   x  =            1      2       gives  2  =      ∑          n      =      N        ∞                                (                            n        N                              )                                1              2        n

Fix N ∈ <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N

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