Exercise.

(a) Explain why, when proving the convergence of a series
$\sum _{n=0}^{\infty}{a}_{n}$ it is permissible to ignore
finitely many terms of the series.

(b) Use (a) together with a form of the comparison test to show
that the series $\sum _{n=0}^{\infty}\frac{1}{(2\sqrt{n}-5)(2\sqrt{n}-3)(2\sqrt{n}-3)}$
converges.

## Solution to selected parts of some of these exercises

### Exercise 1(a)

Let ${s}_{n}=\sum _{k=0}^{n}{a}_{n}$ be the $n$th partial
sum. The series $\sum _{k=0}^{\infty}{a}_{n}$
converges iff the sequence $\left({s}_{n}\right)$ of partial sums converges.

Now suppose that ${t}_{n}=\sum _{k=m}^{n}{a}_{n}$
is the partial sum formed by ignoring the first $m$ entries
and ${t}_{n}\to l\in \mathbb{R}$ as $n\to \infty $.
Then ${s}_{n}={a}_{0}+{a}_{1}+\dots +{a}_{m-1}+{t}_{n}$
and ${s}_{n}\to {a}_{0}+{a}_{1}+\dots +{a}_{m-1}+l$
by continuity of the operation of adding the constant value
${a}_{0}+{a}_{1}+\dots +{a}_{m-1}$. So the
original series also converges.