How do you use the binomial series to expand ${(1 + x)}^{\frac{1}{2}}$ $(1+x)^(1/2)$ ?

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Answer 1 · 2016-06-14T13:29:19

$(1+x)^(1/2)=1+x/2-1/8x^2+1/16x^3-5/128x^4+.......$

According to binomial series $(1+a)^n=1+na+(n(n-1))/(2!)a^2+(n(n-1)(n-2))/(3!)a^3+(n(n-1)(n-2)(n-3))/(4!)a^4+.......$

Hence $(1+x)^(1/2)=1+(1/2)x+((1/2)(1/2-1))/(2!)x^2+((1/2)(1/2-1)(1/2-2))/(3!)x^3+((1/2)(1/2-1)(1/2-2)(1/2-3))/(4!)x^4+.......$

or $(1+x)^(1/2)=1+x/2+((1/2)(-1/2))/(2!)x^2+((1/2)(-1/2)(-3/2))/(3!)x^3+((1/2)(-1/2)(-3/2)(-5/2))/(4!)x^4+.......$

or $(1+x)^(1/2)=1+x/2-1/8x^2+3/48x^3-15/384x^4+.......$

or $(1+x)^(1/2)=1+x/2-1/8x^2+1/16x^3-5/128x^4+.......$

How do you use the binomial series to expand (1+x)^(1/2)(1+x)12(1+x)^(1/2)?