Expand ${{({{x}^{2}}+4)}^{-2}}$ as a series of ascending powers of $x$, up to and including the

Question

Expand ${{({{x}^{2}}+4)}^{-2}}$ as a series of ascending powers of $x$, up to and including the term in ${{x}^{6}}$. State the values of $x$ for which this expansion is valid. Obtain in its simplest form, the term in${{x}^{2n}}$ in this expansion.

Accepted Answer

$(x^2+4)^{-2}=\frac1{16}-\frac{x^2}{32}+\frac{3x^4}{256}-\frac{x^6}{256}+\cdots$. (i) $-2<x<2$. (ii) $\frac{(-1)^n(n+1)x^{2n}}{4^{n+2}}$, $n\ge0$.

Maclaurin Series and Power Series - Binomial Expansion as a Maclaurin Series

Maclaurin Series and Power Series - Binomial Expansion as a Maclaurin Series

Solution:

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