Let P , Q : R → R are polynomials, and Q ( n ) ≠ 0 for n ∈ N Suppose, that deg ⁡ ( P ) &lt; deg ⁡ ( Q ). Prove, that infinite sum ∑ n = 1 ∞ ( − 1 ) n P ( n ) Q ( n )

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Let   P  ,  Q  :      R    →      R   are polynomials, and   Q  (  n  )  ≠  0 for   n  ∈      N   Suppose, that   deg  ⁡  (  P  )  &amp;lt;  deg  ⁡  (  Q  ). Prove, that infinite sum      ∑          n      =      1              ∞        (  −  1      )          n                  P      (      n      )              Q      (      n      )

Trevor Copeland · Accepted Answer

1. We can assume without loss of generality that P and Q are monic. Let p the degree of P, q those of Q (integers, if P=0, the problem is trivial), because the problem of convergence of the series is invariant up to a multiplication by a constant.2. Write   P  (  x  )  =      x    p    +  a      x          p      −      1        +  … and   Q  (  x  )  =      x    q    +  b      x          p      −      1      . We have                                          P            (            n            )                                Q            (            n            )                          −                  n                      p            −            q                                              =                              a                          n                              p                +                q                −                1                                      −            d                          n                              p                +                q                −                1                                      +            …                                n                          2              q                                      ,            so we can find a constant C such that for all n,      |                  P        (        n        )                    Q        (        n        )              −          n              p        −        q              |    ⩽  C      1          n              q        −        p        +        1              ⩽      C          n      2        .3. Now the only task is to show that for   p  ⩾  1, the series      ∑          n      =      1              +      ∞                  (      −      1              )        n                    n      p      is convergent. Dirichlet criterion can be used, or an Abel transform. Let       s    n    :=      ∑          k      =      0        n    (  −  1      )    k  . We have                              ∑                      j            =            n            +            1                                n            +            m                          (        −        1                  )          j                          1                      j            p                                              =                  ∑                      j            =            n            +            1                                n            +            m                          (                  s          j                −                  s                      j            −            1                          )                  1                      j            p                                                            =                  ∑                      j            =            n            +            1                                n            +            m                                    s          j                          1                      j            p                          −                  ∑                      j            =            n                                n            +            m            −            1                                    s          j                          1                      j            p                                                            =                  s                      m            +            n                                    1                      (            m            +            n                          )              p                                      −                  s          n                          1                      n            p                          +                  ∑                      j            =            n            +            1                                n            +            m            −            1                                    s          j                          (                      1                          j              p                                −                      1                          (              j              +              1                              )                p                                              )                .            We conclude using the fact that       |        s    j        |    ⩽  1 for all j

Prove, that infinite sum sum_(n=1)^(infty)(-1)^n(P(n))/(Q(n))

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