is the series is alternating with decreasing terms? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T19:38:21Z http://mathoverflow.net/feeds/question/96252 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/96252/is-the-series-is-alternating-with-decreasing-terms is the series is alternating with decreasing terms? David 2012-05-07T18:56:15Z 2012-05-07T19:35:49Z <p>I am stack on this question. ( I wanted to prove that function, representation of which is exactly this series, has one extremal point). Let <code>$A_j=\{n \in N: n_1+\ldots n_k=n \quad \mbox{and } \quad n_2+2n_3+\ldots +(k+1)n_k=j\}$</code></p> <p>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 $$c_{j}=\sum_{i=0}^j\frac{b_i}{6^{j-i}(j-i)!} \quad \mbox{with} \quad b_j=n!\sum_{n \in A_j}\left(\frac{-1}{n}\right)^j\frac{1}{\prod_{i=1}^k((2i-1)!)^{n_i}n_i!}$$</p>