Answer :
The interval of convergence of the Maclaurin series of (A)-1 is x≈P(x)=x
The concept of Maclaurin series states that the function is just the part of Taylor series of the function if in the Taylor series the center is taken as c = 0 then the Taylor series is called Maclaurin series.
Here we need to find the the interval of convergence of the Maclaurin series.
Let us consider x up to n=3. and the function is x.
Then the Maclaurin series is given by
Find the 1st derivative: f(1)(x)=(f(0)(x))′=(x)′=1
valuate the 1st derivative at the given point: (f(0))′=1
Find the 2nd derivative: f(2)(x)=(f(1)(x))′=(1)′=0
Evaluate the 2nd derivative at the given point: (f(0))′′=0
Find the 3rd derivative: f(3)(x)=(f(2)(x))′=(0)′=0 (steps can be seen here).
Evaluate the 3rd derivative at the given point: (f(0))′′′=0
Now, use the calculated values to get a polynomial:
f(x)≈0/0! x⁰ + 1/1! x¹ + 0/2! x² + 0/3! x³
Finally, after simplifying we get the final answer:
=> f(x)≈P(x)=x
Therefore the Taylor (Maclaurin) series of x up to n=3 is x≈P(x)=x
To know more about Maclaurin series here
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