Hello I was trying to find the coefficient for the member $x^5$ for the expansion: $(1-2x)^{-2}$
Finding the coefficient for binomial expansion
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discrete-mathematics
2 Answers
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Hint: You know the expansion of $\dfrac{1}{1-t}$. Differentiate.
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Around $x=0$, we have that $\dfrac1{1-2x} = 1 + (2x) + (2x)^2 + (2x)^3 + (2x)^4 + (2x)^5 + (2x)^6 + \cdots = \sum_{k=0}^{\infty} 2^kx^k$ Hence, $\dfrac{d}{dx} \left(\dfrac1{1-2x} \right)= \dfrac2{\left(1-2x \right)^2} = \sum_{k=1}^{\infty} k2^k x^{k-1}$ Hence, $\dfrac1{\left(1-2x \right)^2} = \sum_{k=1}^{\infty} k2^{k-1} x^{k-1}$ Now you can read off the power of $x^n$ from above.
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0Thank you very much @Marvis – 2012-10-20