Solved problems on comparison theorem for series

Mika Seppälä: Solved Problems on Comparison Test comparison test Let 0 a k b k for all k.

Mika Seppälä: Solved Problems on Comparison Test 1 OVERVIEW OF PROBLEMS Let 0 a k b k for all k. Assume that the series and both converges. Show that the series converges.

Mika Seppälä: Solved Problems on Comparison Test 2 OVERVIEW OF PROBLEMS Let a k and b k positive for all k. Assume that the series converges and that Show that the series converges.

Mika Seppälä: Solved Problems on Comparison Test OVERVIEW OF PROBLEMS Use Comparison Test to determine whether the series converge or diverge. 3

Mika Seppälä: Solved Problems on Comparison Test Let 0 a k b k for all k. Assume that the series and both converges. Show that the series converges. Problem 1 COMPARISON TEST

Mika Seppälä: Solved Problems on Comparison Test COMPARISON TEST Solution Since converges,. By definition of limit this means, Assume. Since is positive for all k, we have

Mika Seppälä: Solved Problems on Comparison Test COMPARISON TEST Solution (contd) Recall also that by assumptions. Then the Comparison Theorem implies that

Mika Seppälä: Solved Problems on Comparison Test COMPARISON TEST Solution (contd) Remark that is suffices to show that

Mika Seppälä: Solved Problems on Comparison Test Let a k and b k positive for all k. Assume that the series converges and that Show that the series converges. Problem 2 COMPARISON TEST

Mika Seppälä: Solved Problems on Comparison Test COMPARISON TEST Solution Since, there is a number such that Therefore Since, so does

Mika Seppälä: Solved Problems on Comparison Test COMPARISON TEST Solution (contd) This implies by the comparison theorem that

Mika Seppälä: Solved Problems on Comparison Test COMPARISON TEST Problem 3 Solution

Mika Seppälä: Solved Problems on Comparison Test COMPARISON TEST Solution (contd)

Mika Seppälä: Solved Problems on Comparison Test COMPARISON TEST Problem 4 Solution By rewriting,

Mika Seppälä: Solved Problems on Comparison Test COMPARISON TEST Solution (contd) Therefore we can write Hence converges.

Mika Seppälä: Solved Problems on Comparison Test COMPARISON TEST Problem 5 Solution

Mika Seppälä: Solved Problems on Comparison Test COMPARISON TEST Solution (contd)

Mika Seppälä: Solved Problems on Comparison Test COMPARISON TEST Problem 6 Solution

Mika Seppälä: Solved Problems on Comparison Test COMPARISON TEST Solution (contd)

Mika Seppälä: Solved Problems on Comparison Test COMPARISON TEST Problem 7 Solution Since for all n, we obtain

Mika Seppälä: Solved Problems on Comparison Test COMPARISON TEST Solution (contd) We know that the geometric series