TITLE: Struggling to understand a couple of concepts with series
QUESTION [3 upvotes]: I have two questions: neither of which are homework problems but certainly pertain to my ability to do the homework.  The first regards the harmonic series.  The question has been answered often here so I read through those answers.  Although they were helpful, they were similar to what my textbook has.  What confuses me about Nicole Oresme's proof (which is used in my textbook) is why is the "magic" number used for the proof $\frac{1}{2}$?  I get that it works.  I just can't see why it's the number "to beat."
My second question has to do with my textbook's explanation of a power series.  My book has:
A power series about $x=0$ is a series of the form
$$
\sum_{n=0}^{\infty}c_nx^n = c_0+c_1x + c_2x^2+...+c_nx^n +...
$$
A power seies about $x=a$ is a series of the form
$$
\sum_{n=0}^{\infty}c_n(x-a)^n=c_0+c_1(x-a)+c_2(x-a)^2+...+c_n(x-a)^n+...
$$
The book then goes on to say that the first case is a special case taken when $a=0$.  Ok, that makes sense to me.  However, if, as stated in the first case, $x=0$ why doesn't every term starting with $c_1$ collapse to 0?  Then, in the second case, if $x=a$, why don't they also collapse to 0 starting with $c_1$?  The only reasonable conclusion that I can find is that $x$ seems to refer to two different things.  This seems a poor use of notation to me, but could someone please clear up my confusion?
Thanks, Andy

REPLY [1 votes]: In the first case, it we aren't talking about the series when or at $x = 0$, it's a series "about"/"around" the point $x = 0$, i.e. near the point $x = 0$. Similarly, "about the point $x = a$" is the series around or near* where $x = a$.
For example, in Wikipedia, you'll find the following definition, which is much less ambiguous, I think, than what you've been given to work with:

In mathematics, a power series (in one variable) is an infinite series of the form
$$f(x) = \sum_{n=0}^\infty a_n \left( x-c \right)^n = a_0 + a_1 (x-c)^1 + a_2 (x-c)^2 + a_3 (x-c)^3 + \cdots$$
where $a_n$ represents the coefficient of the nth term, $c$ is a constant, and $x$ varies around $c$ (for this reason one sometimes speaks of the series as being centered at $c$)...In many situations c is equal to zero...[boldface mine].

I think you're correct that the definitions you're given seems to confuse the general term $x$ with the value at which the series is centered: $c$, i.e., seeming as though $x$ is used to "refer to two different things", and is perhaps a poor use of notation, in that respect.
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