The Binomial expansion of given expression, (1+x)⁻¹ is 1 − x + x² - x³+ x⁴ + ...
Binomial theorem primarily helps to find the expanded value of the algebraic expression of the form (x + y)ⁿ
Using the Binomial theorem we can expand the series of ( x + y )ⁿ as
( x + y)ⁿ = ∑ ⁿCₖ xⁿ⁻ᵏ yᵏ
k= 0 to n
= xⁿ + (n(n-1)/2!)x⁽ⁿ⁻¹⁾y + (n(n-1)(n-2)/3!)x⁽ⁿ⁻²⁾y² + .......
where n ≥ 0 and (ⁿCₖ ) is known as a binomial coefficient and they should be positive integers.
We have given that (1+x)⁻¹ comparing it with ( x + y )ⁿ we get, y = 1 ; n = -1
Binomial expansion for this is followed the same way as expanding larger power expansion,
So (1+x)⁻¹ = 1 + (-1)x + (-1)(-1–1)/2! x² + (-1)(-1–1)(-1–2)/3! x³+…….
= 1 -x + x²- x³ +….
but this will result in an infinite series and will converge for values belonging to
|x|<1. So, the required binomial series is
1 -x + x² - x³ +….
To learn more about binomial theorem, refer:
https://brainly.com/question/13672615
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