I would like some help in calculating the following sum, I don't know how to handle double factorial.
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1The summation variable has no relation to the summand. See this question for something similar. https://math.stackexchange.com/questions/2370144/peculiar-sum-regarding-the-reciprocal-binomial-coefficients/2370344#2370344 – Donald Splutterwit Oct 01 '17 at 12:27
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2You should include the actual question in the body not just in the title, you should also include your own work/thoughts on the problem. – Henrik supports the community Oct 01 '17 at 12:32
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1As @Henrik notes, your question should be clear without the title. After the title has drawn someone's attention to the question by giving a good description, its purpose is done. The title is not the first sentence of your question, so make sure that the question body does not rely on specific information in the title. – Simply Beautiful Art Oct 01 '17 at 12:40
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Also, please follow the guidelines outlined by How to ask a good question? and How to ask a homework question?. Low quality questions run the risk of being closed and deleted, and repeated closures and deletions may trigger a question ban. – Simply Beautiful Art Oct 01 '17 at 12:41
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@SimplyBeautifulArt Thanks for the help. – Aco Oct 01 '17 at 12:45
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Don't ask for "help", meaning assistance but not doing work for you, unless that's what you mean. When you're ready to include context, ideally your efforts, only then ask for help. Or ask for a hint. In the case of your question, you are really asking "I want someone to find the sum for me,..." which is not okay on MSE. – amWhy Oct 01 '17 at 15:36
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The $n$-th term is $$(-1)^n\frac{1\times 3\times5\times\cdots\times(2n-1)} {2\times 4\times6\times\cdots\times(2n)} =\frac{1}{n!}\left(-\frac12\right)\left(-\frac32\right)\left(-\frac52\right)\cdots\left(-\frac{2n-1}2\right).$$ According to the binomial theorem, that is the coefficient of $x^n$ in the power series for $(1+x)^{-1/2}$, etc.
Angina Seng
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