Question
find the first four terms in the binomial expansion of 1/1+x
Answers
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GPT 3.5
To find the first four terms in the binomial expansion of 1/(1+x), we can use the binomial theorem.
The binomial theorem states that for any real number x and any positive integer n, the expansion of (1 + x)^n can be found using the formula:
(1 + x)^n = 1 + nx + (n(n-1)x^2)/2! + (n(n-1)(n-2)x^3)/3! + ...
In this case, we are looking for the expansion of 1/(1 + x), which is the same as expanding (1 + x)^(-1).
Using the formula for the first four terms:
(1 + x)^(-1) = 1 - x + x^2 - x^3 + ...
Therefore, the first four terms in the expansion of 1/(1 + x) are:
1, -x, x^2, -x^3
The binomial theorem states that for any real number x and any positive integer n, the expansion of (1 + x)^n can be found using the formula:
(1 + x)^n = 1 + nx + (n(n-1)x^2)/2! + (n(n-1)(n-2)x^3)/3! + ...
In this case, we are looking for the expansion of 1/(1 + x), which is the same as expanding (1 + x)^(-1).
Using the formula for the first four terms:
(1 + x)^(-1) = 1 - x + x^2 - x^3 + ...
Therefore, the first four terms in the expansion of 1/(1 + x) are:
1, -x, x^2, -x^3