1. ## Taylor series and radius of convergence

I have a couple of interrelated questions about series:

Although I've used series for some time, I never really understood what the range of convergence really means. If I have a Taylor series of the function Exp(x) around x=0, does that mean that if I take into consideration only the first 3 terms of the series, it will be very close to the original function around x=0, and that the more terms I take into consideration of the series, the polynomial generated by the series will be getting closer and closer to the real function, both around x=0 and also farer and farer away from that point?

This question is related too to asking how does a graphic calculator calculate what sin(0.5) or cos(0.75) values are. Could anyone explain? I'd like something more visual or not too deep on maths, if possible.

Also, what does it mean to say that the series is centered around x=0, or x=5?

2. Originally Posted by devouredelysium
I have a couple of interrelated questions about series:

Although I've used series for some time, I never really understood what the range of convergence really means. If I have a Taylor series of the function Exp(x) around x=0, does that mean that if I take into consideration only the first 3 terms of the series, it will be very close to the original function around x=0, and that the more terms I take into consideration of the series, the polynomial generated by the series will be getting closer and closer to the real function, both around x=0 and also farer and farer away from that point?
whether taking 3 terms give a "very close" result depends upon how fast the series converges, not on the radius of convergence.
The radius of convergence simply means that the series does converges (no matter how slowly) inside that radius and does not converge outside it.

This question is related too to asking how does a graphic calculator calculate what sin(0.5) or cos(0.75) values are. Could anyone explain? I'd like something more visual or not too deep on maths, if possible.
I don't see how it could be. A calculator does NOT use Taylor's series to calculate trig or exponential functions. See Internal Programming ofthe 9100A Calculator

Also, what does it mean to say that the series is centered around x=0, or x=5?
Every power series is of the form $\displaystyle \sum_{n=0}^\infty a_n(x- x_0)^n$. It is "centered around" $\displaystyle x_0$.

3. Thanks for the answer. I am right now reading the link you provided. My question about the series being centered about x0 was more about what is its meaning? What does it mean for a function to be centered around 5? I know I will have all my terms in the form (x-x0)^n, but what is the difference in having it centered around 0 or 5?