Question

Using the equations frac{1}{1-x}=1+x+x^{2}+x^{3}+ldots and the fact that frac{2}{1+3 x}=2 frac{1}{1-(-3 x)} we have: frac{2}{1+3 x}=2-3 x+9 x^{2}-27 x^{3}+ldots frac{2}{1+3 x}=1-3 x+9 x^{2}-27 x^{3}+ldots frac{2}{1+3 x}=2-6 x+18 x^{2}-54 x^{3}+ldots frac{2}{1+3 x}=1+3 x+9 x^{2}+27 x^{3}+ldots frac{2}{1+3 x}=2+6 x+18 x^{2}+54 x^{3}+ldots

          Using the equations
frac{1}{1-x}=1+x+x^{2}+x^{3}+ldots and the fact that frac{2}{1+3 x}=2 frac{1}{1-(-3 x)} we have:
frac{2}{1+3 x}=2-3 x+9 x^{2}-27 x^{3}+ldots
frac{2}{1+3 x}=1-3 x+9 x^{2}-27 x^{3}+ldots
frac{2}{1+3 x}=2-6 x+18 x^{2}-54 x^{3}+ldots
frac{2}{1+3 x}=1+3 x+9 x^{2}+27 x^{3}+ldots
frac{2}{1+3 x}=2+6 x+18 x^{2}+54 x^{3}+ldots

$Using the equations frac11-x=1+x+x^2+x^3+ldots and the fact that frac21+3 x=2 frac11-(-3 x) we have: frac21+3 x=2-3 x+9 x^2-27 x^3+ldots frac21+3 x=1-3 x+9 x^2-27 x^3+ldots frac21+3 x=2-6 x+18 x^2-54 x^3+ldots frac21+3 x=1+3 x+9 x^2+27 x^3+ldots frac21+3 x=2+6 x+18 x^2+54 x^3+ldots$

Added by Dan

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