Final answer:
To find the Taylor series expansions at x=0 for the given functions, we use the general formula of the Taylor series and apply known series expansions, modifying them to fit the particular function.
Step-by-step explanation:
The question involves finding the Taylor series expansions at x=0 for a set of functions listed in the exercises. Taylor series is a way to represent a function as an infinite sum of terms calculated from the values of its derivatives at a single point. To find these series, we typically use the formula:
Taylor series of f(x) around x=a is given by:
f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + f'''(a)(x-a)^3/3! + ...
For example, to find the Taylor series for the function f(x) = 1/(1-2x) at x=0, we start by noting that this is a geometric series with a first term of 1 and a common ratio of 2x:
1 + 2x + (2x)^2 + (2x)^3 + ...
This approach applies to each function in the exercise list. The key to finding the correct Taylor series expansion for each function is to recognize known series expansions such as the binomial theorem, the expansions of e^x, sin(x), and cos(x), then adjust them according to the specific form of the function in question