Why Taylor series does not converge for all x in the domain of the function

Question

Example: $$ f(x)=\frac{1}{1+x} \qquad x\neq-1 $$ $$ f(x)=1-x+x^2-x^3+x^4-x^5+\;... \qquad |x| < 1 $$

Why Taylor series does not converge for all x in the domain of the function?

Answer 1

This series is obtained from Newton's generalization of the binomial theorem which is only applicable for $|x|\lt 1$. This can be also obtained from the infinte geometric series, as

$$\lim_{n\to\infty}1-x+x^2-\cdots+(-1)^nx^n = \lim_{n\to\infty}\frac{1-x^{n+1}}{1+x}$$

Clearly, this only converges if $|x|\lt1$

Although it is useful to note, that Gottfried Leibniz admitted that the series,

$$1-2+4-8+\cdots = ?$$

Could either yield positive or negative infinty depending on how you subtract, and therefore neither can be correct, and the actual value should be finite. To quote:

"Now normally nature chooses the middle if neither of the two is permitted, or rather if it cannot be determined which of the two is permitted, and the whole is equal to a finite quantity."

He associated it with $\frac13$ which now happens to be the accepted convergent representation for this divergent sum, and can be otained by substituting $x=2$ in the Taylor series for $\frac{1}{1+x}$. So this formula isn't entirely inapplicable. Citation

Answer 2

A quick answer is imagine $|x| > 1$, then $x^n \to \pm \infty$ as $n\to \infty$.

So your series is adding and subtracting larger and larger numbers at every step. How could this possibly converge?

A more mathematical reason is that this is the same as a well known formula : $$\dfrac{1}{1-x} = 1+x+x^2+x^3+...\qquad |x| < 1$$ multiplying both sides by $1-x$ you obtain : $$1 = (1+x+x^2+x^3+...+x^n)(1-x) = 1-x+x-x^2+x^2 - ... -x^{n-1}+x^{n+1} - x^n$$ Most terms cancel and you get : $$1 =\lim_{n\to\infty} 1-x^n$$ This only holds for $|x|<1$.

Answer 3

It's actually surprising that the Taylor series converges to the function anywhere other than $x=0$ -- functions that are equal to their Taylor series are actually rather rare.

Of course, most of the functions people actually study do have this property of being "analytic", so it's easy not to get a sense of how special this property is.

It turns out there is a pleasing geometric description of what the radius of convergence of a Taylor series will be: if you look at the complex plane, your analytic function may have some singularities. The Taylor series around any point will converge in a disk around that point whose radius is as large as it can possibly be without including any of those singularities.

Your particular $f$ has a singularity at $x = -1$, and that is its only singularity in the complex plane. Thus, the Taylor series around $x=0$ will have a radius of convergence equal to $1$. Its Taylor series around $ x = 41$ will have a radius of converge equal to $42$. Its Taylor series around $x = \mathbf{i}$ will have radius of convergence $\sqrt{2}$, and so forth.