problemas de treinamento para o Putnam (solu,c~oes)

["Luis Lopes" <llopes@xxxxxxxxxxxxx>]

Sauda,c~oes,

Mando as solu,c~oes para os dois problemas que mandei
para a lista na semana passada.

[ ]'s
Lu'is

1)  We use the ratio test.  The $n$-th term of the series (starting
with $n = 0$) is $n! (19/7)^n /(n+1)^n$.  By a simple calculation,
\[
\frac{a_n}{a_{n-1}} = \frac{19/7}{(1 + 1/n)^n} \rightarrow
\frac{19}{7e} \quad \text{as} \quad n \rightarrow \infty.
\]
Since $19/7 = 2.\overline{714285} < 2.71828182846\ldots = e$, the
series converges by the ratio test. 

2)  Let $c_n = a_n -a_{n+1}$ so $b_n = c_n - c_{n+1}$.
By the given conditions, both $(a_n)$ and $(c_n)$ decrease to zero.
By telescoping sums,
\[
\sum_{k=m}^n c_k = a_m - a_{n+1} \qquad \text{and} \qquad
\sum_{k=m}^n b_k = c_m - c_{n+1}
\]
so
\[
\sum_{k=m}^{\infty} c_k = a_m \qquad \text{and} \qquad
\sum_{k=m}^{\infty} b_k = c_m.
\]
Since the series in question have nonnegative terms, we may
reverse the order of summation to obtain
\[
\sum_{n=1}^{\infty} n b_n = \sum_{n=1}^{\infty} b_n \sum_{k=1}^n 1
= \sum_{k=1}^{\infty} \sum_{n=k}^{\infty} b_n = 
\sum_{k=1}^{\infty} c_k = a_1.
\]


-----Mensagem Original----- 
De: Luis Lopes <llopes@ensrbr.com.br>
Para: <obm-l@mat.puc-rio.br>
Enviada em: Sexta-feira, 15 de Setembro de 2000 15:27
Assunto: problemas de treinamento para o Putnam


> Sauda,c~oes,
> 
> O professor Cecil Rousseau me mandou uma lista de
> problemas de treinamento para o Putnam. Minha
> inten,c~ao e' colocar 2 problemas/semana na lista
> e na semana seguinte suas solu,c~oes.
> 
> Os problemas est~ao escritos em ingl^es e em
> LaTeX.
> 
> Ai' v~ao os dois primeiros:
> 
> 1) Is the following series convergent or divergent?
> \[
> 1 + \frac{1}{2}  \left(  \frac{19}{7} \right) + \frac{2!}{3^2}
> \left(  \frac{19}{7} \right)^2 + \frac{3!}{4^3} \left(  \frac{19}{7}
> \right)^3 + \frac{4!}{5^4}  \left( \frac{19}{7} \right)^4 + \cdots \; .
> \]
> \hspace*{\fill} (A-3, 1942)
> 
> 2) Let $\{a_n\}$ be a decreasing sequence of positive numbers
> with limit 0 such that
> \[
> b_n = a_n - 2a_{n+1} + a_{n+2} \geq 0
> \]
> for all $n$.  Prove that
> \[
> \sum_{n=1}^{\infty} n b_n = a_1.
> \]
> \hspace*{\fill} (A-3, 1948)
> 
> N~ao sei o significado de (A-3, 1942) e (A-3, 1948). Vou me
> informar.
> 
> [ ]'s
> Lu'is
> 
>