# Sequences and series: exercise sheet 8

Exercise.

(a) Explain why, when proving the convergence of a series $∑ n = 0 ∞ a n$ 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 $∑ n = 0 ∞ 1 ( 2 n - 5 ) ( 2 n - 3 ) ( 2 n - 3 )$ converges.

## Solution to selected parts of some of these exercises

### Exercise 1(a)

Let $s n = ∑ k = 0 n a n$ be the $n$th partial sum. The series $∑ k = 0 ∞ a n$ converges iff the sequence $( s n )$ of partial sums converges.

Now suppose that $t n = ∑ k = m n a n$ is the partial sum formed by ignoring the first $m$ entries and $t n → l ∈ ℝ$ as $n → ∞$. Then $s n = a 0 + a 1 + … + a m - 1 + t n$ and $s n → a 0 + a 1 + … + a m - 1 + l$ by continuity of the operation of adding the constant value $a 0 + a 1 + … + a m - 1$. So the original series also converges.