2
$\begingroup$

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.

$\endgroup$
  • 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
1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.