Question Type 10: Using binomial theorem for negative indices Exercises

Use the first four terms of the expansion of $(1 + x)^{-3}$ to approximate $(1.1)^{-3}$.

Find the first four terms of the series expansion of $(1 + x)^{-2}$.

Approximate $(4 + 0.2)^{-2}$ using the first three nonzero terms of the binomial expansion of $(4 + x)^{-2}$, taking $x=0.2$.

Determine the coefficient of $x^5$ in the expansion of $(1 + 3x)^{-4}$.

Provide a general formula for the coefficient of $x^k$ in the expansion of $(a + x)^{-m}$, where $m$ is a positive integer.

Expand $(2 + x)^{-1}$ in ascending powers of $x$ up to and including $x^3$, and state the interval of convergence.

Expand $(5 - 2x)^{-1}$ in ascending powers of $x$ up to and including the term in $x^3$.

Expand $(3 - x)^{-3}$ in ascending powers of $x$ up to and including the term in $x^3$.

Find the coefficient of $x^3$ in the expansion of $(1 + 2x)^{-2}$.