Next page : Mclaurin series - a first take
A power series is defined as
where the coefficients, ak, are constants. If all these constants above some numbers of terms are 0 then we have a finite power series, else we have an infinite power series. Finite power series are polynomials .
If this limit exists then we say that the series converges to that limit, else we say that it diverges.
An infinite geometric series can be seen as a power series. If we have
then the first term u1 = a and the common ratio is r=bx. The sum of an infinite geometric series is
I.e. it converges when r is between –1 and 1. So for our series we get
If we have a=1 and b=1 we get
This we could prove by by the standard proof for the sum of a geometric series. We could also have found that by doing a polynomial division. We have that
The above is not a rigours proof, but it at least gives us a hint of that it is correct.
In the figure below the function and the series for six and 50 terms are plotted. We can clearly see that the series coverages to the function as values of x somewhat between –1 and 1.Up a level : Power Series
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