Series Convergence: Ignoring a Finite Number of Terms

A common question we will ask is the following: given an infinite series $\sum_{n=1}^{\infty} a_{n}$, determine whether that series converges or not. So far, we only know of one general type of series for which we can answer that question: geometric series. (Recall that we can actually do better with geometric series - not only can we determine convergence or divergence, we can also calculate the value of a convergent geometric series.)

Suppose we are given a series $\sum a_{n}$ which is not quite a geometric series, but becomes a geometric series for sufficiently large $n$. That is, we can find a (possibly large) integer $N$ such that $a_{n} = ar^{n}$ for all $n > N$. We don't have a general formula for $a_{n}$ when $n \leq N$. Nevertheless, we can still easily determine whether or not $\sum a_{n}$ converges or not: we simply check whether $|r| < 1$. The values of $a_{n}$ for small $n$ don't affect whether or not the partial sums $s_{n}$ converge, because their contribution to the partial sums $s_{n}$ is a constant finite value (namely, $a_{1} + \cdots + a_{N}$).

As a matter of fact, we never used the fact that $a_{n}$ eventually became geometric. More generally, if $\sum a_{n}$ is a convergent series, we can add a finite number of terms to the beginning of this series, or remove a finite number of terms from the beginning of the series, without affecting the convergence of the series. For example, if $\sum a_{n}$ converges, then so does

$$10 + 10^{10} + 10^{10^{10}} + 10^{10^{10^{10}}} + \sum a_{n}.$$

Similarly, if $\sum a_{n}$ converges, then so does

$$a_{1000} + a_{1001} + a_{1002} + \cdots = \sum_{n=1000}^{\infty} a_{n}.$$

Exercise 1. Show that analogous statements hold for divergent series: that is, if $\sum a_{n}$ is divergent, then adding or removing a finite number of terms from the beginning of this series still results in a divergent series.

Notice, however, that the fact that we are only adding finitely many terms to the series is essential. That is,

Exercise 2. Give an example of a convergent series $\sum a_{n}$ and an infinite sequence $b_{n}$ such that $\sum (a_{n} + b_{n})$ diverges.

Exercise 3. Give an example of a divergent series $\sum a_{n}$ and an infinite sequence $b_{n}$ such that the sum $\sum (a_{n} + b_{n})$ converges.

When it comes to determining convergence and divergence of a series $\sum a_{n}$, we see that only the asymptotic behavior of $\sum a_{n}$ matters: that is, only the behavior of the sequence $a_{n}$ as $n \rightarrow \infty$ affects convergence or divergence. We will frequently use this observation in future sections to simplify and assist in testing for convergence of series.

Adding, changing, or removing finitely many summands of an infinite series doesn't affect the convergence/divergence of the series. For example, $0.5^{-2016} + 2017^{2017} + 0.5^3 + 0.5^4 + 0.5^5 + \dots$ still converges, because it only differs from the convergent infinite geometric series $1 + 0.5 + 0.5^2 + 0.5^3 + 0.5^4 + 0.5^5 + \dots$ in finitely many terms.