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I am reading Stephen Abbott's book: "Understanding Analysis" (Second Edition) ...
I am focused on Chapter 6: Sequences and Series of Functions ... and in particular on power series ...
I need some help to understand Theorem 6.5.1 ... specifically, some remarks that Abbott makes after the proof of the theorem ...
Theorem 6.5.1 and Abbott's remarks read as follows:View attachment 8585In the above text by Abbott (after the proof ... ) we read the following:
" ... ... The main implication of Theorem 6.5.1 is that the set of points for which a given power series converges must necessarily be $$\{ 0 \} , \mathbb{R} $$, or a bounded interval centred around $$x = 0$$ ... ... "I was wondering why the above quote would be true ...
My thinking is that since $$\mid \frac{x}{x_0} \mid \lt 1$$ we have that $$-x_0 \lt x \lt x_0$$ ... ...
If $$x_0$$ was simply $$0$$ ( ... and there were no other points where the power series converged) then $$\{ 0 \}$$ would be the set of points for which the power series converged ...
If $$x_0 = b$$, say, then the power series would converge in the bounded interval $$( -b, b )$$ ...
Is that correct so far?BUT ... how does Abbott arrive at the fact that an implication of the above theorem is that the power series may converge on all of $$\mathbb{R}$$ ...Hope that someone can help ...
Peter
I am focused on Chapter 6: Sequences and Series of Functions ... and in particular on power series ...
I need some help to understand Theorem 6.5.1 ... specifically, some remarks that Abbott makes after the proof of the theorem ...
Theorem 6.5.1 and Abbott's remarks read as follows:View attachment 8585In the above text by Abbott (after the proof ... ) we read the following:
" ... ... The main implication of Theorem 6.5.1 is that the set of points for which a given power series converges must necessarily be $$\{ 0 \} , \mathbb{R} $$, or a bounded interval centred around $$x = 0$$ ... ... "I was wondering why the above quote would be true ...
My thinking is that since $$\mid \frac{x}{x_0} \mid \lt 1$$ we have that $$-x_0 \lt x \lt x_0$$ ... ...
If $$x_0$$ was simply $$0$$ ( ... and there were no other points where the power series converged) then $$\{ 0 \}$$ would be the set of points for which the power series converged ...
If $$x_0 = b$$, say, then the power series would converge in the bounded interval $$( -b, b )$$ ...
Is that correct so far?BUT ... how does Abbott arrive at the fact that an implication of the above theorem is that the power series may converge on all of $$\mathbb{R}$$ ...Hope that someone can help ...
Peter