Tail of Convergent Series tends to Zero

Theorem
Let $$\sum_{n=1}^\infty a_n$$ be a convergent series.

Let $$N \in \mathbb{N}^*$$ be a natural number.

The expression $$\sum_{n=N}^\infty a_n$$ is known as the tail of the series $$\sum_{n=1}^\infty a_n$$.

Then:
 * $$\sum_{n=N}^\infty a_n$$ is convergent;
 * $$\sum_{n=N}^\infty a_n \to 0$$ as $$N \to \infty$$.

That is, the tail of a convergent series tends to zero.

Proof
Let $$\left \langle s_n \right \rangle$$ be the sequence of partial sums of $$\sum_{n=1}^\infty a_n$$.

Let $$\left \langle s'_n \right \rangle$$ be the sequence of partial sums of $$\sum_{n=N}^\infty a_n$$.


 * We are going to show that $$\left \langle s'_n \right \rangle$$ fulfils the Cauchy criterion.

That is, for any $$\forall \epsilon > 0: \exists N: \forall m, n > N: \left|{s'_n - s'_m}\right| < \epsilon$$.

So, let $$\epsilon > 0$$.

As $$\left \langle s_n \right \rangle$$ is convergent, it conforms to the Cauchy criterion by Convergent Sequence is Cauchy Sequence.

Thus $$\exists N: \forall m, n > N: \left|{s_n - s_m}\right| < \epsilon$$.

Now $$s_n = \sum_{k=1}^n a_k = \sum_{k=1}^{N-1} a_k + \sum_{k=N}^n a_k = s_{N-1} + s'_n$$ and similarly $$s_m = s_{N-1} + s'_m$$.

Thus $$s'_n = s_n - s_{N-1}, s'_m = s_m - s_{N-1}$$.

So:

$$ $$ $$ $$ $$

So $$\sum_{n=N}^\infty a_n$$ fulfils the Cauchy criterion, and by Convergent Sequence is Cauchy Sequence is convergent.


 * Now we show that $$\sum_{n=N}^\infty a_n \to 0$$ as $$N \to \infty$$.

We know that $$\left \langle s_n \right \rangle$$ is convergent. Let its limit be $$l$$.

Thus we have $$l = \sum_{n=1}^\infty a_n = s_{N-1} + \sum_{n=N}^\infty a_n$$.

So $$\sum_{n=N}^\infty a_n = l - s_{N-1}$$.

But $$s_{N-1} \to l$$ as $$N-1 \to \infty$$ and the result follows.