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

Blaine Ortega 2022-08-13 Answered
Let P , Q : R R are polynomials, and Q ( n ) 0 for n N Suppose, that deg ( P ) < deg ( Q ). Prove, that infinite sum
n = 1 ( 1 ) n P ( n ) Q ( n )
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Answers (1)

Trevor Copeland
Answered 2022-08-14 Author has 21 answers
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
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