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 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.