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In the Taylor series expansion of ex about x = 2, the coefficient of (x - 2)4 is 
a) 1/4!  
b) 24/4!
c) e4/4!   
d) e2/4!  
Correct answer is option 'D'. Can you explain this answer?
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In the Taylor series expansion of ex about x = 2, the coefficient of (...
To find the coefficient of (x - 2)^4 in the Taylor series expansion of ex about x = 2, we can use the formula for the nth derivative of ex:

f(n)(x) = en(x - a)n

where f(n)(x) is the nth derivative of f(x), a is the point about which we are expanding the series (in this case, a = 2), and n is the order of the derivative.

In the case of the Taylor series expansion of ex about x = 2, we want to find the coefficient of (x - 2)^4, which means we need to find the 4th derivative of ex and evaluate it at x = 2.

1. Find the 4th derivative of ex:
f(4)(x) = e^x

2. Substitute x = 2 into the 4th derivative:
f(4)(2) = e^2

3. Divide the result by 4! to find the coefficient of (x - 2)^4:
Coefficient = e^2 / 4!

Therefore, the coefficient of (x - 2)^4 in the Taylor series expansion of ex about x = 2 is e^2 / 4!.

In the given options, option 'D' (e^2 / 4!) is the correct answer.
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In the Taylor series expansion of ex about x = 2, the coefficient of (x - 2)4 isa)1/4! b) 24/4!c) e4/4! d) e2/4!Correct answer is option 'D'. Can you explain this answer?
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