Binomial series

$(a+x)^n = a^n + {n \choose 1}a^{n-1}x^1 + {n \choose 2}a^{n-2}x^2 + {n \choose 3}a^{n-3}x^3 +...$

Special cases are:
$(a+x)^{1} = a^{1}+x^{1}$
$(a+x)^{2} = a^{2}+2a^{1}x^{1}+x^{2}$
$(a+x)^{3} = a^{3}+3a^{2}x^{1}+3a^{1}x^{2}+x^{3}$
$(a+x)^{4} = a^{4}+4a^{3}x^{1}+4a^{2}x^{2}+4a^{1}x^3+x^{4}$

These expansions below are valid in the interval: $-1

$(a+x)^{-1} = 1-x+x^2-x^3+x^4-...$
$(a+x)^{-2} = 1-2x+3x^2-4x^3+5x^4-...$
$(a+x)^{-3} = 1-3x+6x^2-10x^3+15x^4-...$

And these expansions (below) are valid in the interval: $-1

$(a+x)^{-1/2} = 1-\frac{1}{2}x+\frac{1\cdot3}{2\cdot4}x^{2}-\frac{1\cdot3\cdot5}{2\cdot4\cdot6}x^{3}+...$

$(a+x)^{1/2} = 1+\frac{1}{2}x-\frac{1}{2\cdot4}x^{2}+\frac{1\cdot3}{2\cdot4\cdot6}x^{3}-...$

$(a+x)^{-1/3} = 1-\frac{1}{3}x+\frac{1\cdot4}{3\cdot6}x^{2}-\frac{1\cdot4\cdot7}{3\cdot6\cdot9}x^{3}+...$

$(a+x)^{1/3} = 1+\frac{1}{3}x-\frac{2}{3\cdot6}x^{2}+\frac{2\cdot5}{3\cdot6\cdot9}x^{3}-...$

All of these can be found in Schaum's outlines, Mathematical Handbook of Formulas and Tables, Third Edition, Murray R. Spiegel PhD, Seymour Lipschutz PhD, John Liu PhD, page 138.

These expansions also work for triangular matrices with ones in the main diagonal by replacing $a$ with the identity matrix $I$ and replacing $x$ with $X$ which stands for the part of the lower triangular matrix below the main diagonal.

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