Lot bn be a bounded sequence of nonnegative numbers and 0 le

Lot {b_n} be a bounded sequence of nonnegative numbers and 0 lessthanorequalto r

Solution

For example, let b_n = (1/rⁿ).

It is clear that the b_n are nonnegative and satisfy the hypotheses of statement.

Yet, s_n = b₁r¹ + b₁r¹ + b₁r¹... + b_nrⁿ

Putting value of b_n in s_n, we get

s_n=1+1+1+...n times = n

which shows s_n converges.