Tail of Convergent Series tends to Zero

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

Let $N \in \N_{\ge 1}$ be a natural number.

Let $\displaystyle \sum_{n \mathop = N}^\infty a_n$ be the tail of the series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$.

Then:
 * $\displaystyle \sum_{n \mathop = N}^\infty a_n$ is convergent
 * $\displaystyle \sum_{n \mathop = N}^\infty a_n \to 0$ 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 $\displaystyle \sum_{n \mathop = 1}^\infty a_n$.

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

It will be shown that $\left \langle s'_n \right \rangle$ fulfils the Cauchy criterion.

That is:
 * $\forall \epsilon \in \R_{>0}: \exists N: \forall m, n > N: \left\vert{s'_n - s'_m}\right\vert < \epsilon$

Let $\epsilon \in \R_{>0}$ be a strictly positive real number.

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

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

Now:
 * $\displaystyle s_n = \sum_{k \mathop = 1}^n a_k = \sum_{k \mathop = 1}^{N - 1} a_k + \sum_{k \mathop = N}^n a_k = s_{N - 1} + s'_n$

and similarly:
 * $s_m = s_{N - 1} + s'_m$

Thus:
 * $s'_n = s_n - s_{N - 1}$

and:
 * $s'_m = s_m - s_{N - 1}$

So:

So $\displaystyle \sum_{n \mathop = N}^\infty a_n$ fulfils the Cauchy criterion.

By Convergent Sequence is Cauchy Sequence it follows that it is convergent.

Now it is shown that $\displaystyle \sum_{n \mathop = N}^\infty a_n \to 0$ as $N \to \infty$.

We have that $\left \langle s_n \right \rangle$ is convergent,

Let its limit be $l$.

Thus we have:
 * $\displaystyle l = \sum_{n \mathop = 1}^\infty a_n = s_{N - 1} + \sum_{n \mathop = N}^\infty a_n$

So:
 * $\displaystyle \sum_{n \mathop = N}^\infty a_n = l - s_{N - 1}$

But $s_{N - 1} \to l$ as $N - 1 \to \infty$.

The result follows.

Also see

 * Convergent Sequence with Finite Number of Terms Deleted is Convergent