We can solve this problem using the definition of
Binomial theorem for a rational index.
Binomial Theorem for a rational Index
If n is a nonzero rational number and –1 < x < 1 then
(1+x)n =
1+x+x2 +
x3 + ... +
xr + ... to ∞
In the above expansion, the first term must be unity. In the expansion of
(a+x)n , where n is either a negative integer or a fraction, we proceed as follows
(a+x)n =
[a(1+)]n =
an(1+)n =
an[1+n+()2+...] By using this definition we can find the power of x is valid
We have
=
=
The expansion in powers of x is valid if
|| < 1 i.e., - 2 < x < 2
(value inside the square root should be positive)
Therefore
The expansion of
in powers of x is valid if
- 2 < x < 2