Answer by Gary for Summation of a difference of two square-roots
I shall derive a complete asymptotic expansion for the sum as $n\to+\infty$. Note that for $|x|<1$,$$\sqrt {1 + x} - \sqrt {1 - x} - x = \sum\limits_{m = 1}^\infty \binom{4m}{2m} \frac{{x^{2m + 1}...
View ArticleAnswer by Greg Martin for Summation of a difference of two square-roots
Since\begin{align*}\sqrt{1+\frac1k} - \sqrt{1-\frac1k} &= \frac{(\sqrt{1+\frac1k} - \sqrt{1-\frac1k})(\sqrt{1+\frac1k} + \sqrt{1-\frac1k})}{\sqrt{1+\frac1k} + \sqrt{1-\frac1k}} \\&=...
View ArticleSummation of a difference of two square-roots
Does a closed form expression exist for$$\sum_{k=1}^{n}\left(\sqrt{1+\frac{1}{k}}-\sqrt{1-\frac{1}{k}}\right)$$I obtained$$0.97423066\ln(n)+1.2019463$$using log regression and it works very well for my...
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