two examples in analysis I want to ask for two examples in the following cases:
1) Given a bounded sequence $\{a_n\}$, $$\lim_{n\to \infty}{(a_{n+1}-a_n)}=0$$ but $\{a_n\}$ diverges.
2) A function defined on real-line $f(x)$'s Taylor series converges at a point $x_0$ but does not equal to $f(x_0)$.
Thanks for your help.
Edit
in 2), I was thinking of the Taylor series of the function $f$ at the point $x_0$.

For 1):
$$
1,\frac12,0,\frac13,\frac23,1,\frac34,\frac24,\frac14,0,\frac15,\frac25,\frac35,\frac45,1,\frac56,\frac46,\frac36,\frac26,\frac16,0,\frac17,\dots
$$
I leave it to you to find an explicit formula for $a_n$.
