Advertisements
Advertisements
Question
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)`
Advertisements
Solution
`(5 + x^2)^(2/3) = {5(1 +x^2/5)}^(2/3)`
= `5^(2/3) [(1 + x^2/5)^(2/3)]`
= `5^(2/3) {1 + 2/3(x^2/5) + (2/3(2/3 - 1))/(2.1) (x^2/5)^2 ...}`
= `5^(2/3) {1 + (2x^2)/15 - 2/(9 xx 2) (x^4/25) ...}`
= `5^(2/3) {1 + (2x^2)/15 - x^4/225 ...}`
Hence `|x^2/5| < 1`
⇒ |x2| < 5
APPEARS IN
RELATED QUESTIONS
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
`(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"^(-2x)`
Write the first 6 terms of the exponential series
`"e"^(1/2x)`
If p − q is small compared to either p or q, then show `root("n")("p"/"q")` ∼ `(("n" + 1)"p" + ("n" - 1)"q")/(("n"- 1)"p" +("n" + 1)"q")`. Hence find `root(8)(15/16)`
Find the coefficient of x4 in the expansion `(3 - 4x + x^2)/"e"^(2x)`
Find the value of `sum_("n" = 1)^oo 1/(2"n" - 1) (1/(9^("n" - 1)) + 1/(9^(2"n"- 1)))`
Choose the correct alternative:
The coefficient of x6 in (2 + 2x)10 is
Choose the correct alternative:
The coefficient of x8y12 in the expansion of (2x + 3y)20 is
Choose the correct alternative:
If a is the arithmetic mean and g is the geometric mean of two numbers, then
Choose the correct alternative:
The sum of an infinite GP is 18. If the first term is 6, the common ratio is
Choose the correct alternative:
The coefficient of x5 in the series e-2x is
Choose the correct alternative:
The value of `1/(2!) + 1/(4!) + 1/(6!) + ...` is
Choose the correct alternative:
The value of `1 - 1/2(2/3) + 1/3(2/3)^2 1/4(2/3)^3 + ...` is
