Tail of Convergent Series tends to Zero

Theorem
Let $\sequence {a_n}_{n \mathop \ge 1}$ be a sequence of real numbers.

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

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

Proof
Let $\sequence {s_n}$ be the sequence of partial sums of $\ds \sum_{n \mathop = 1}^\infty a_n$.

Let $\sequence {s'_n}$ be the sequence of partial sums of $\ds \sum_{n \mathop = N}^\infty a_n$.

It will be shown that $\sequence {s'_n}$ fulfils the Cauchy criterion.

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

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

As $\sequence {s_n}$ is convergent, it conforms to the Cauchy criterion by Convergent Sequence is Cauchy Sequence.

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

Now:

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

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

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

We have that $\sequence {s_n}$ is convergent,

Let its limit be $l$.

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

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

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