# is the series is alternating with decreasing terms? [closed]

I am stack on this question. Let $A_j=\{n \in N: n_1+\ldots n_k=n \quad \mbox{and } \quad n_2+2n_3+\ldots +(k+1)n_k=j\}$

Help me please to show if the following series is alternating with decreasing terms? $$S=c_0+\frac{c_1}{n}+\sum_{i=2}^{\infty}\frac{c_i}{n^i},$$ where $$c_{j}=\sum_{i=0}^j\frac{b_i}{6^{j-i}(j-i)!} \quad \mbox{with} \quad b_j=n!\sum_{n \in A_j}\left(\frac{-1}{n}\right)^j\frac{1}{\prod_{i=1}^k((2i-1)!)^{n_i}n_i!}$$

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What is the motivation for this calculation? –  Antonio Vargas May 6 '12 at 19:15
The rison is that I wanted to proof that function, representation of which is exactly this series, has one extremal point. –  Alex K. May 6 '12 at 22:41
Unfortunately, the definition of $A_j$ makes no sense. Presumably, $A_j\subseteq\mathbb N$ but then, what are the $n_i$s? –  Did May 23 '12 at 8:23