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A well-behaved function can be expanded into a power series. This means that for all non-negative integers there are real numbers such that

Let us calculate the first four derivatives using :

Setting equal to zero, we obtain

Let us write for the -th derivative of  We also write — think of as the "zeroth derivative" of  We thus arrive at the general result where the factorial  is defined as equal to 1 for and and as the product of all natural numbers for Expressing the coefficients in terms of the derivatives of at we obtain

This is the Taylor series for 

A remarkable result: if you know the value of a well-behaved function and the values of all of its derivatives at the single point then you know at all points  Besides, there is nothing special about so is also determined by its value and the values of its derivatives at any other point :


cos Edit












Some basic checking:



arctan Edit




 . See  .

Second derivative  Edit




Third derivative  Edit





If you continue to calculate derivatives, you will produce the following sequence:


















Some basic checking:



arcsin Edit





Simple differential equations eliminate the square root and make calculations so much easier.


Then   where   and  

Differentiating both sides:






Differentiating both sides:




When   Calculation of more derivatives yields:











and so on.





As programming algorithm:Edit


As implemented in Python:Edit

from decimal import * # Default precision is 28.

π = ("3.14159265358979323846264338327950288419716939937510582097494459230781")
π = Decimal(π)

x = Decimal(2).sqrt()/2 # Expecting result of π/4

xSQ = x*x
X = x*xSQ

top = Decimal(1)
bottom = Decimal(2)

bottom1 = bottom*3
sum = x + X*top / bottom1

status = 1
for n in range(5,200,2) :
    X = X*xSQ
    top = top*(n-2)
    bottom = bottom*(n-1)
    bottom1 = bottom*n
    added = X*top/bottom1
    if (added < 1e-29) :
        status = 0
    sum += added

if status :
    print ('error. count expired.')
else :
    print (x, sum==π/4, n)
0.707106781186547524400844362 True 171

In practiceEdit

If   is close to   the calculation of   will take forever.

If you limit   to   then   and each term is guaranteed to be less than half the preceding term.

If   let  


External linksEdit