Working with binomial series Use properties of power series, substitution, and factoring to find the first four nonzero terms of the Maclaurin series for the following functions. Use the Maclaurin series

\((1+x)^{-2}=1-2x+3x^2-4x^3+..., for -1

\((1+4x)^{-2}\)

The Maclaurin series is \((1+x)^{-2}=1-2x+3x^2-4x^3+...\)

Now obtain the series \((1+4x)^{-2}\) by substituting x=4x in the series of \((1+x)^{-2}=1-2x+3x^2-4x^3+...\) and simplify.

Therefore,

\((1+4x)^{-2}=1-2(4x)+3(4x)^2-4(4x)^3+...\)

\(=1-8x+3(16)x^2-4(64)x^3+...\)

\(=1-8x+3(16)x^2-4(64)x^3+...\)

Thus, the maclurin series (1+4x)^{-2}=1-8x+48x^2-256x^3+...\)

\((1+x)^{-2}=1-2x+3x^2-4x^3+..., for -1

\((1+4x)^{-2}\)

The Maclaurin series is \((1+x)^{-2}=1-2x+3x^2-4x^3+...\)

Now obtain the series \((1+4x)^{-2}\) by substituting x=4x in the series of \((1+x)^{-2}=1-2x+3x^2-4x^3+...\) and simplify.

Therefore,

\((1+4x)^{-2}=1-2(4x)+3(4x)^2-4(4x)^3+...\)

\(=1-8x+3(16)x^2-4(64)x^3+...\)

\(=1-8x+3(16)x^2-4(64)x^3+...\)

Thus, the maclurin series (1+4x)^{-2}=1-8x+48x^2-256x^3+...\)