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algorithmsoup.wordpress.com | ||
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mikespivey.wordpress.com
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| | | | | A few months ago Mathematics Magazine published a paper of mine, "A Combinatorial View of Sums of Powers." In it I give a combinatorial interpretation for the power sum $latex \sum_{k=1}^n k^m$, together with combinatorial proofs of two formulas for this power sum. (An earlier version of some of the results in this paper actually... | |
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ckrao.wordpress.com
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| | | | | In this post I would like to prove the following identity, motivated by this tweet. $latex \displaystyle n! \prod_{k=0}^n \frac{1}{x+k} = \frac{1}{x\binom{x+n}{n}} = \sum_{k=0}^n \frac{(-1)^k \binom{n}{k}}{x+k}$ The first of these equalities is straightforward by the definition of binomial coefficients. To prove the second, we make use of partial fractions. We write the expansion $latex \displaystyle... | |
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hbfs.wordpress.com
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| | | | | $latex n!$ (and its logarithm) keep showing up in the analysis of algorithm. Unfortunately, it's very often unwieldy, and we use approximations of $latex n!$ (or $latex \log n!$) to simplify things. Let's examine a few! First, we have the most known of these approximations, the famous "Stirling formula": $latex \displaystyle n!=\sqrt{2\pi{}n}\left(\frac{n}{e}\right)^n\left(1+\frac{1}{12n}+\frac{1}{288n^2}-\frac{139}{51840n^3}-\cdots\right)$, Where the terms... | |
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willsrandomweirdness.wordpress.com
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| | | Reblogged on WordPress.com | ||
