Sequences and series: exercise sheet 8


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

Solution to selected parts of some of these exercises

Exercise 1(a)

Let = =0 be the th partial sum. The series =0 converges iff the sequence ( ) of partial sums converges.

Now suppose that = = is the partial sum formed by ignoring the first entries and as . Then =0+1++ -1+ and 0+1++ -1+ by continuity of the operation of adding the constant value 0+1++ -1 . So the original series also converges.