A theorem on the convergence of series

Question

consider a sequence of non-negative numbers like $ \{a_1, a_2,...\} $ with the following property: $$\sum_{i=1}^{\infty} a_i=b$$ Is there any theorem which states the following:
for any c>0 there exists $k \in Z^+$ such that $$\sum_{i=k}^{\infty} a_i<c$$

Answer 1

Let $b_k:=\sum_{i=k}^{\infty} a_i$. Since $\sum_{i=1}^{\infty} a_i$ is convergent, we have $ b_k \to 0$.

Answer 2

Yes; this follows from the definition of series convergence as convergence of the partial sums.

Since the series converges to $b$, we know that the sequence of partial sums $\sum_{i=1}^n a_i$ ($ n \in \mathbb Z$) converges to $b$ as $n \to \infty$. Thus given $c>0$, we can find $n \in \mathbb Z$ such that $|b - \sum_{i=1}^n a_i| <c$. But since $b = \sum_{i=1}^\infty a_i$, $b - \sum_{i=1}^n a_i = \sum_{i=n+1}^\infty a_i$; thus $|\sum_{i=n+1}^\infty a_i|<c$. Taking $k=n+1$ gives your result.

Answer 3

If you consider the sequence of partial sums $c_n:=\sum^{n}_{k=1}a_i$, then we know by the Cauchy convergence criterion that the difference between consecutive terms gets arbitrarily small, which is essentially what you're asking.

Answer 4

$$ \sum_{i=k}^\infty a_i= \sum_{i=0}^\infty a_i - \sum_{i=0}^{k-1}a_i= b- \sum_{i=0}^{k-1} a_i\to 0. $$