Can you approximate 127 without using a calculator Use taylo

Can you approximate 1/27 without using a calculator? Use taylor expansion up to first derivative.

Solution

he taylor series expansion of 1/x centered at c = 1 can be written as:

f(c)/0! + f\'(c)(x - c)/1! + f\'\'(c)(x - c)^2 / 2! + f\'\'\'(c)(x - c)^3 / 3! + ... + f^n(c)(x - c)^n / n!

Under sigma notation, this is represented as:

Sum(from n = 0 ? ?) [ f^n(c) (x - c)^n / n! ]

Where f^n(c) denotes the nth derivative of f at c.

First we find a general expression for the nth derivative of f, and then use that to find the nth coefficient in the series.

f(x) = 1/x ? f(1)/0! = 1
f\'(x) = -1/x^2 ? f\'(1)/1! = -1
f\'\'(x) = 2/x^3 ? f\'\'(1)/2! = 1
f\'\'\'(x) = -6/x^4 ? f\'\'\'(1)/3! = -1

At this point we can see that f^n(x) = (-1)^n n! / x^(n+1) and f^n(c) / n! = (-1)^n * n! / n! = (-1)^n

put x=27

we get 1/27=0.037042