# Is the Weierstrass function given in Counterexamples in Analysis a typo?

Let $0 < a < 1$; let $b$ be an odd integer; let $ab > 1 + \frac{3\pi}{2}$; let $f: x \mapsto \sum_{n \geq 0}a^{n}\cos (b^{n}\pi x): \Bbb{R} \to \Bbb{R}$. Then $f$ is everywhere continuous and nowhere differentiable.

On page 39 of Counterexamples in Analysis (Dover edition), the authors write the Weierstrass function $f$ as $f(x) = \sum_{n \geq 0}b^{n}\cos (a^{n} \pi x)$; then I do not see how it can converge (of this "typo" series)...

If this is really an obvious typo, then there should already be someone that pointed it out; but so far I did not find any source doing this, so I ask here to clarify this point.

• It's weird. In the index it's stated that there's an errata, but it's not included somehow.
– user99914
Nov 1 '15 at 5:47
• @JohnMa Thank you for feedback. Maybe I should contact the authors (if they are still alive. :)) Nov 1 '15 at 5:52

The typo is the constraints on $a$ and $b$. In the book it says that $b$ is an odd integer, it should actually be $a$. And $b$ should be $0<b<1$. That, or the $a$ and $b$ should be switched in the series.

To see this converges, just use the Weierstrass M-test. $|a^n\cos(b^n \pi x)|\leq a^n$ and $\sum a^n$ converges.

• Thanks for feedback. The remedy I guess is like yours: $a$ should be put where $b$ is and $b$ should be put where $a$ is. Nov 1 '15 at 5:50
• @GudsonChou If you look at en.wikipedia.org/wiki/Weierstrass_function#Construction you'll see they have the same definition with $a$ and $b$ switched.
– user223391
Nov 1 '15 at 5:51
• Oh, right, thanks for that. Now I tend to maintain more that it is simply a typo. Nov 1 '15 at 5:54
• @GudsonChou Actually the proof it converges is really straightforward. Just use Weierstrass M-test.
– user223391
Nov 1 '15 at 5:58
• Sorry? Yes, it is clear that it converges; but this is not my question :) Nov 1 '15 at 5:59