Infinite Sequences and Series

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

`1/(5 + x)`

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

`2/(3 + 4x)^2`

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

`(5 + x^2)^(2/3)`

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

`(x + 2) - 2/3`

Find `root(3)(10001)` approximately (two decimal places

Prove that `root(3)(x^3 + 6) - root(3)(x^3 + 3)` is approximately equal to `1/x^2` when x is sufficiently large

Prove that `sqrt((1 - x)/(1 + x))` is approximately euqal to `1 - x + x^2/2` when x is very small

Write the first 6 terms of the exponential series
e^5x