Expand the following in ascending powers of $x$ and find the condition on $x$ for which the binomial expansion is valid.

$(x+2{)}^{-\frac{2}{3}}$

Write the first $6$ terms of the exponential series

$e^{5x}$

Write the first $6$ terms of the exponential series

$e^{-2x}$

Write the first $6$ terms of the exponential series

$e^{\frac{1}{2}x}$

Write the first $4$ terms of the logarithmic series $\log (1+4x)$.Find the intervals on which the expansions are valid.

Write the first $4$ terms of the logarithmic series $\log (1-2x)$.Find the intervals on which the expansions are valid.