Well, maybe the above is a bad example because they are equal iff -1<x<1.
Hmm...maybe I am some what understanding it....
As you may have guessed, I am currently studying cal II, the section about: sequences, series, power series, et al...
I do not understand the need for representing functions as power series? I know how to mechanically find the answer (following procedure basically) but I lack the understanding behind it.
Can someone please explain it to me with some clear examples.
So what if I knew that ?
Is that saying:
When x = 3 then:
I am lost....
No "what if" about it. You don't know that- it's not true! What is true is that provided that -1< x< 1
That's a very good reason for learning about power series- so you won't make foolish mistakes like that!Is that saying:
When x = 3 then:
I am lost....
Another reason for writing functions in terms of power series is to be able to extend the functions to other "number systems".
For example, we know how to add, subtract, multiply and divide complex number and matrices but what is or cos(z) or sin(z) if z is a complex number or a matrix?
Answer: write those functions as power series and use the properties of sums and products. That's why analytic functions are so important in Complex Analysis- they are the functions that can be written as power series!
A popular misconception but it is not usual for a calculator to use a power series to evaluate trig functions (or if a series is involved it is a very minor part of the algorithm, and then it is more likely to be a polynomial approximation than a power series if you see the distinction).
CB