Hi, I have the following problem with finding a power series representation of a

Question

Hi, I have the following problem with finding a power series representation of a function.

1) Why do we always put x=0 to find C? (Picture 1 and 2)

2) Why can we put -(1/5)C1 into a new variable C and stil become the same answer? (Picture 2)

V EXAMPLE 7 Find a power series representation for for tan SOLUTION We observe that f'(x) 1/(1 x2) and find the required series by integrating the power series for 1/(1 x2) found in Example 1. x2 x dx tan x 1 x x x x C x To find C We put x 0 and obtain C tan '0 0. Therefore 2n+ 1 tan x x 2n 1 3 5 7 Since the radius of convergence of the series for 1/(1 x2) is 1, the radius of conver- gence of this series for tan x is also 1.

Explanation / Answer

a) at only x= 0, we get the finite value as RHS contains infinite number of terms which becomes finite only at x = 0 otherwise we cannot get the value of c.

b) we can replace -1/5c1 by c as both will give arbitrary value such as if we take c1 = 5, then -1/5c1 = -1 = c (say)

Like this we are replacing the above i.e, whatever value we get from -1/5c1, we are taking that value equal to c.

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