How can I understand this? − q ∑ n = 0 ∞ t p + n q p + n q = ∑ n = 0 q − 1 ω − n p ln ⁡ ( 1 − ω n t )

Question

How can I understand this?  −  q      ∑          n      =      0              ∞                  t              p        +        n        q                    p      +      n      q        =      ∑          n      =      0              q      −      1            ω          −      n      p        ln  ⁡  (  1  −      ω          n        t  )

Austin Solis · Accepted Answer

I&#039;m proving this as an equality of power series in the indeterminate t, where   log  ⁡  (  1  −  x  ) is defined by       ∑          k      ⩾      0        −            x      k        k  . Here goes (throwing out a minus sign):      ∑          n      =      0              q      −      1            ω          −      n      p            ∑          k      ⩾      0                  (              ω        n            t              )        k              k    =      ∑          n      =      0              q      −      1            ∑          k      ⩾      0                          ω                  n          (          k          −          p          )                            t        k              k    =      ∑          k      ⩾      0                  t      k        k        ∑          n      =      0              q      −      1            ω          n      (      k      −      p      )      If   k  −  p  ≡  0    mod      q      ∑    0          q      −      1            ω          n      (      k      −      p      )        =      ∑    0          q      −      1        1  =  q. If   k  −  p  ≢  0    mod      q, then write   x  =      ω          k      −      p        =  x, another nontrivial       q          t      h       root of unity. We have  (  1  +  x  +  …  +      x          q      −      1        )  (  x  −  1  )  =      x    q    −  1  ⇒  (  1  +  x  +  …  +      x          q      −      1        )  =                    x        q            −      1              x      −      1        =            1      −      1              x      −      1        =  0Thus this equals  q      ∑          k      ≡      p            mod                  q                  t      k        k  which isn&#039;t quite what you have, unless   p  &amp;lt;  q

How can I understand this? − q <munderover> ∑ <mro

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New Questions in Calculus 2