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Struggling with understanding this taylor vs. maclaurin series stuff.

So a few questions. Let's say that we have some function f(x).

1. By saying that we want to find the power series of f(x) and nothing else, are we implicitly stating that we are looking for a maclaurin series or a taylor series? OR do we have to specify around what point we're looking for this power series?

2. I went on wolfram alpha and I found the taylor series of sin(x) at x =2 and the maclaurin series of sin(x) (at x = 0). and then I evaluated the answers at the same x value (x = 4). And I got different answers. I thought they should come out to the same value since we're still expanding the same initial function, sin(x). Ultimately, I'm finding it difficult to understand how these two seemingly different power series are converging to one function.

3. What does it mean for a power series to be centered around a value, other than for it to be the center of the circle of convergence.

Hope these questions made sense. I just want to get a really strong intuition for maclaurin vs. taylor series.

Thanks!

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# Another maclaurin vs. taylor series question

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