Answer to Question #1751 in Calculus for Imee Carla

Question #1751
How can I use the concept in differentials to approximate a certain number which is not a perfect square?
Expert's answer
You can expand the function of square root in a series :
We can represent the number as a sum of nearest perfect square root R and the difference of the number and this sq.root (N-R), it can be both positive and negative.
f(x) = √N = √(R + (N-R)) = √R √(1 + (N-R)/R).

Denote (N-R)/R as α , |α|<1. The function √(1 +α) can be represented as a sum:
√(1 +α)= 1 + 1/2 α - 1/8 α2 + 1/16 α3 - 5/128 α4 + ... This expression was obtained by expanding the function √(1 +α) in a Teylor series in neiborhood of 1. √(1 +α) = sum from n=0 to infinity ( f(n)(1)/n! αn ), where f(n) (1) the n-th derivative of the function f(a)=√x at the point x = 1.
Thus the final fomula would be as
√N = √R (1 + 1/2 α - 1/8 α2 + 1/16 α3 - 5/128 α4 +...), where α = (N-R)/R

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