Why does the taylor series of $\ln (1 + x)$ only approximate it for $-1<x \le 1$?

The selected answer to the above question says that for a a power series, the interval of convergence for the Taylor series is syymetric around $a$, where $a$ is the domain value the power series is centered around.

Could someone provide an intuitive reason as to why this is true? Note I am not just looking for a formal proof, I am looking for intuition.

Thanks for the help.

  • 1
    $\begingroup$ For reasons apparent when considering complex arguments, power series converge inside a circle around the center (that's why they talk about "radius of convergence"). This circle (centered on the real line) gives a range symmetric around it in the reals. $\endgroup$ – vonbrand Apr 20 '13 at 21:04
  • $\begingroup$ @vonbrand: you mean, inside a disk around the center. $\endgroup$ – xyzzyz Apr 20 '13 at 21:05
  • $\begingroup$ What's the difference, @xyzzyz? $\endgroup$ – anon Apr 20 '13 at 21:05
  • $\begingroup$ @xyzzyz "Inside a circle" (on a disk) is correct, while "inside a disk" (what is the inside of a disk, if not the disk itself?) is disputable. $\endgroup$ – Pedro Tamaroff Apr 20 '13 at 22:56
  • $\begingroup$ "Inside a circle" can be interpreted as "on a circle". The most unambiguous ways of expressing it is I thing "inside a disk bounded by a circle". $\endgroup$ – xyzzyz Apr 20 '13 at 23:55

Hmm, tricky one.

The intuition of the formal proof (not sure if this counts!) is that if a series converges conditionally, then it is at worst 'balanced on the edge' of convergence (root test gives 1); hence if you slightly improve convergence by decreasing $|x|$, the series converges 'properly' (root test gives something proportionally less than 1). Hence

the regions of conditional convergence are at most points balanced precariously on a boundary of a region of absolute convergence, which obviously must be symmetric.

Thus $(-1,1]$ type intervals are as asymmetric as you can get. This generalizes immediately to the complex plane, where the absolutely convergent region is trivially circular, and the conditionally convergent region is a part of the circle bounding this disc.

Morally, therefore, the symmetric nature ultimately arises from the simple fact that

the series at $+x$ converges absolutely if and only if it does at $-x$

combined with the above observation that conditionally convergent regions are 'narrow', 'unstable', 'borderline cases', etc.


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