Prove that lim α → 0 + ∑ n = 1 ∞ ( − 1 ) n − 1 n α = 1 / 2

Question

Prove that       lim          α      →              0        +                  ∑          n      =      1              ∞                  (      −      1              )                  n          −          1                            n              α              =  1      /    2

sleuteleni7 · Accepted Answer

∑          n      =      1              ∞                  (      −      1              )                  n          −          1                            n              α              =      ∑          m      ≥      1            ∫          2      m      −      1              2      m        α      x          −      α      −      1        d  x  =      1    2    +      α    2        ∑          m      ≥      1        (      ∫          2      m      −      1              2      m            x          −      α      −      1        d  x  −      ∫          2      m              2      m      +      1            x          −      α      −      1        d  x  )It suffices to prove that the latter series is bounded on   [  0  ,  1  ]

Prove that <munder> <mo movablelimits="true" form="prefix">lim <mrow class="MJX-TeXAtom-O

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