We know the following theorem:
Theorem (Hardy-Littlewood). Let $(b_n)$ be a real sequence such that
(i) $(nb_n)$ is bounded,
(ii) $\displaystyle\lim_{x\to 1^-} \sum_{n=0}^\infty b_n x^n=\ell$.
Then $\sum b_n$ converges and $\displaystyle\sum_{n=0}^\infty b_n=\ell$.
The question : I would like to know some examples of applications of this theorem, and some counter-examples when you change the hypothesis.
What I did.
- So far I have found that if you omit hypothesis (i), then $b_n=(-1)^n$ is a counter-example, because
$$\lim_{x\to 1^-} \sum_{n=0}^\infty (-x)^n=\lim_{x\to 1^-}\frac1{1+x}=\frac 12$$
but
$$\sum (-1)^n$$
does not converges.
- I also found an application with $b_n=(-1)^{n+1}/n$ which verifies (i) trivially, and (ii) also because
$$\sum_{n=0}^\infty \frac{(-1)^{n+1}}n x^n=\log(1+x)\xrightarrow[x\to 1^-]{}\log 2$$
and you can deduce from that the sum of the altering harmonic series:
$$\sum_{n=0}^\infty \frac{(-1)^{n+1}}n=\log(2).$$
Edit
I just found out that this theorem is still true when you replace (i) by :
$$(i')\quad \exists m,\quad\forall n,\quad nb_n\geqslant m.$$
Is there an example of an interesting application when $(b_n)$ verifies (i') but not (i)?