Series convergence/divergence question

Question

Suppose        and        for all n. Of the series, which will definitely converge and which will definitely diverge? Explain.d will definitely converge and which will definitely diverge? Explain.which will definitely converge and which will definitely diverge? Explain.           which will definitely converge and which will definitely diverge? Explain.

Answer 1

0 ≤ b_n ≤ 2ⁿ ≤ a_n

0 ≤ c_n ≤ 2^-n ≤ d_n

∑ 2ⁿ diverges (Geometric Series with r > 1)

So, by the Comparison Test, ∑ a_n diverges

∑ 2^-n is a convergent Geometric Series with r = 1/2.

So, by the Comparison Test, ∑ c_n converges

There is not enough information to draw any conclusions for ∑ b_n and ∑ d_n