Power series and the value of the expression $0^0$ I have a doubt regarding the value of the expression $0^0$. I know this value is taken as indeterminate as far as limits are concerned. All was fine upto now. But when I encountered power series, I found out when $x=a$ in the expression summation $[b (x-a)^n]$ where $n=0$ to infinity, of the power series, then the series always converges which is understood. But what bothers me is its value converges to $b$ and not $0$. That is the first term of the power series is written as $b \cdot 0^0$ and $0^0$ is taken as $1$ and not as indeterminate. 
Can anyone tell me why this is so? How is it possible at one time we define $0^0$ as indeterminate and at other time its value is taken as $1$? Could anyone help me on this one? Thanks.
 A: Generally $0^0$ is taken to be $1$, so that the term of degree $0$ in a polynomial or power series can be written as $c_0x^0$, rather than having some special exception for $x=0$. It makes the notation cleaner.
As an aside, when one talks about an expression being an indeterminant form, say  $E(a,b)$, one usually means that if we have functions $f(x),g(x)$ such that $\lim\limits_{x\to x_0}f(x)=a$ and $\lim\limits_{x\to x_0} g(x)=b$, we cannot conclude that $\lim\limits_{x\to x_0}E(f(x),g(x))=E(a,b)$. In the case of $0^0$, this means that even if $\lim\limits_{x\to x_0}f(x)=0$ and $\lim\limits_{x\to x_0} g(x)=0$ we cannot conclude that $\lim\limits_{x\to x_0}f(x)^{g(x)}=0^0$, no matter how $0^0$ is defined. Note that this is not the same as saying that $0^0$ is undefined. It just says the expression does not "play well" with taking limits.
A: As a limit, 0^0 is and remains an indeterminate form. There is no answer to 0^0 in terms of a number. However, in order to make certain formulas "work" the convention is to "let" 0^0 to be equal to 1. While those formulas were never designed to find answers to 0^0, they happen to work only if we accept 0^0 equals 1. In series we find this situation, with the Binomials distribution, this situation also pops around the corner. Look, sometimes we just need to be "flexible" and do certain "modifications" to make applications come out the right way. It is not something that should bother you. As a real number, 0^0 does not exist, in limit form, it's called an indeterminate form, and once in a while we bend (not "break") the rules :)
