Sequences and series: exercise sheet 8

Exercise.

(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.