Power Series

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Power Series
Power Series
Power series?


Ok. I know how to the Interval of Convergence of the power series. But when I check for the endpoints. Is it true that if it is centered at 0, 1 end converges and the other diverges. And if it is set at =/ 0 then either both converge or both diverge? If that is not the case isn't there some rule like that to determine?

No, there's no rule. Whenever you are finding the interval of convergence for a power series, you have to simply plug the endpoint values into the series and simplify, then decide whether that simplified series converges using any suitable test.



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Power Series - Finding the Interval of Convergence

use the binomial series to expand the function as a power series?


(1-x)^2/3

The main thing that I can't find any information on is that if k is positive is it still k(k-n)/n! it seems like it should be (k+n)n! neither of the 2 examples in my text book use a positive k and I can't find any positive k examples online. Also I don't really know how to go from binomial notation to power series notation, thus making it hard to find a radius of convergence. thanks

The binomial theorem says

(1-x)^(2/3) = [1^(2/3)]/0! + [(2/3)(1^(-1/3))(-x)]/1! + [(2/3)(-1/3)(1^(-4/3))(-x)²]/2! + [(2/3)(-1/3)(-4/3)(1^(-7/3))(-x)³]/3! + ...
= 1 - (2/3)x - (1/9)x² - (4/81)x³ - ...

It is, as you know, an infinite series.

By trial and error I found that the coefficient of x^n (for n > 1) is

  n
- Π (3j - 5) / [(3^n)(n!)]
 j=1

where Π denotes product.

This should be provable by induction.

Then the Ratio Test has

|a_(n+1) / a_n| = [3(n+1) - 5] | x | / [3(n+1)]

The limit on this as n→∞ is less than 1 for |x| < 1. You'll need to check x = ±1

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