Question

How do you use the binomial series to expand `(x+1)^20`?

Answer 1

`(x+1)^20 = sum_(k=0)^20 ((20),(k)) x^(20-k)`

Explanation:

By the binomial theorem:

`(a+b)^n = sum_(k=0)^n ((n),(k)) a^(n-k) b^k`

where `((n),(k)) = (n!)/((n-k)! k!)`

Putting `a=x`, `b=1` and `n=20` we get:

`(x+1)^20 = sum_(k=0)^20 ((20),(k)) x^(20-k)`

`color(white)((x+1)^20) = ((20),(0)) x^20 + ((20),(1)) x^19 + ... + ((20),(20))`

If you just want a single term from this expansion, then it is probably easiest to calculate directly.

For example, the fourth term is:

`((20),(3)) x^17 = (20!)/(17! 3!) x^17 = (20xx19xx18)/(3xx2xx1) x^17 = 1140 x^17`

If you want to write out every term, then it may be easier to write out Pascal's triangle as far as the `21`st row - hoping that you make no errors on the way. The `21`st row starts `1, 20, 190, 1140,...` consisting of the values:

`((20),(0)), ((20),(1)),...,((20),(20))`

conveniently providing all the coefficients you want.