1. Prove that if Σ|a n | converges and (b n ) is a bounded sequence, then Σa n b n converges. 2. Suppose that (a n ) is a sequence of positive numbers. For each n ∈ N, let b n = (a 1 + a 2 + … + a n...

1. Prove that if Σ|a_n| converges and (b_n) is a bounded sequence, then Σa_nb_n
converges.

2. Suppose that (a_n) is a sequence of positive numbers. For each n ∈ N, let b_n
= (a₁
+ a₂
+ … + a_n)/n. Prove that Σb_n
diverges to + ∞.

3. Mark each statement True or False. Justify each answer.

(a) If Σa_n
converges and 0 ≤ b_n
≤ an, then Σb_n
converges.

(b) If Σ|a_n| diverges, then Σa_n
is conditionally convergent.

(c) Changing the first few terms in a series may affect the value of the sum of the series.

(d) Changing the first few terms in a series may affect whether or not the series converges.