Sure, let's find Power Series representations of these functions one by one.
1. For the first function, f(x) = 1 - x/4. Its Power-Series representation is as straightforward as the function itself, which is: 1 - x/4. In other words, if we put this function in the power series representation, it does not change.
2. f(x) = 1 + 4x/x^2. From this function, we can simplify this to f(x) = 1 + 4/x. Thus, our Power-Series representation reflects this simplified equation.
3. Up next we have f(x) = (3x^2 + 2x - 1)/(7x - 1). To transform this into a power series, we need to perform polynomial division. Doing the division and expressing it as a series gives us: 1 + 5x + 32x^2 + 224x^3 + 1568x^4 + 10976x^5 + O(x^6). This is our Power-Series representation. The O(x^6) indicates that the series continues and the terms beginning with x^6 have not been computed.
4. The final function is f(x) = ln(x^2 + 1). This logarithmic function can be expressed as a Power Series after using a Maclaurin series expansion. In this case, the series becomes: x^2 - x^4/2 + O(x^6).
Remember that when we say O(x^n), we mean that the terms starting from the nth term in the series have not been computed. This is an important notation in Power Series.
And there we have it! We've found Power Series representations for each of your functions!