Asking if an infinite series will converge or diverge

Question

I have two questions regarding the convergence of infinite series. One, if $\sum_{n=1}^{\infty}a_n$ converges to L, where L is defined, does $a_n < L$ for each n?

Two, if $a_n$ is defined for each n, and $\sum_{n=j}^{\infty}a_n$ converges for some integer j, then does $\sum_{n=j}^{\infty}a_n$ converge for all j?

My response: for one, this is a no for me, since you can make $a_n=0$, which will promptly converge to 0, but $0 \not < 0$. For two, I think this is based right off of one of the properties of series, where if a portion of the series converges, then the entire series converges, right? Correct me if I'm wrong (:

Answer 1

You are more or less correct for both questions, but we can expand on this a little bit.

1. Note that if $\sum_{n=1}^{\infty}a_{n}$ converges, then $\lim_{n\to\infty}a_{n}=0$. Now, consider $\sum_{n=1}^{\infty}\frac{-1}{n^{2}}=-\frac{\pi^{2}}{6}=L$. It follows that $a_{n}>L$ for sufficiently large $n$ since $L$ is negative and $a_{n}$ must eventually be getting very close to zero.
2. Your intuition is essentially right here. For example, if $\sum_{j=1}^{\infty}a_{j}$ converges and you ignore finitely-many terms of the series, then the remaining sum still converges.