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 $x_0$.

2) Define the function $f$ as follows:  $$f(x)=e^{-1/x^2}\ \text{if } x>0$$ and $$f(x)=0\ \text{if } x\leq 0.$$  Now consider the Taylor series centered at zero.  This provides an example of a non-analytic $C^\infty$ function.  This taylor series converges everywhere, but is identically zero, and $f(x)$ is not identically zero.
Hope that helps,
