1. Suppose that (an) is a sequence of numbers such that for all n, | a n+1 − a n | ≤ b n , where Σb n is convergent. Show that (an) converges. 2. Show that the series diverges. Why doesn’t this...

1. Suppose that (an) is a sequence of numbers such that for all n, | a_n+1
− a_n
| ≤ b_n, where Σb_n
is convergent. Show that (an) converges.

2. Show that the series

diverges. Why doesn’t this contradict the alternating series test?

3. Let (a_n) be a decreasing sequence of positive numbers such that lim a_n
= 0. Show that the sum s of the alternating series Σ(−1)ⁿ⁺¹a_n
lies between any pair of successive partial sums. That is, show that s_2n
≤ s ≤ s_2n+1. Then use this to conclude that, for all n, | s – s_n
| ≤ a_n+1. Thus the error made in stopping at the nth term does not exceed the absolute value of the next term.