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If there any way to expand the following? $$\left(\sum_{i=1}^nx_i\right)^{\frac{1}{2}}$$ and more generally, a way to expand $$\left(\sum_{i=1}^nx_i\right)^{\frac{p}{q}}$$ where $gcd(p,q) = 1$ More closed to my original problem, is there any formula for: $$\left(\sum_{i=1}^nx_i\right)^{\frac{1}{2}} - \sum_{i=1}^nx_i^{\frac{1}{2}}$$ |
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I think, you can use binomial theorem for rational exponents inductively for expanding $(\sum_{i=1}^{n}x_i)^\frac{1}{2}$. I am not very sure whether one can use multinomial theorem for rational exponents. When you expand $(1+x)^{\frac{1}{2}}$, It has infinitely many terms, so to your other question, It is very less likely to have any formula, as one of the term in the expansion has infinitely many terms while $\sum_{i=1}^{n}x_i^{\frac{1}{2}}$ has only finitely many. |
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