Explain why or why not
Determine whether the following statements are true and give an explanation or counterexample.
f.If the sequence {aₙ} diverges, then the sequence {0.000001aₙ} diverges.

13–52. Limits of sequences

Find the limit of the following sequences or determine that the sequence diverges.

{bₙ}, where

bₙ = { n / (n + 1)if n ≤ 5000

ne⁻ⁿif n > 5000 }

Explain why or why not

Determine whether the following statements are true and give an explanation or counterexample.

c.The convergent sequences {aₙ} and {bₙ} differ in their first 100 terms, but aₙ = bₙ for n > 100.

It follows that limₙ→∞aₙ = limₙ→∞bₙ.

Explain why or why not

Determine whether the following statements are true and give an explanation or counterexample.

a.The terms of the sequence {aₙ} increase in magnitude, so the limit of the sequence does not exist.

Sequences versus series

a.Find the limit of the sequence { (−⁴⁄₅)ᵏ }.

Find two different explicit formulas for the sequence {1, -2, 3, -4, -5 .....}

27–34. Working with sequences Several terms of a sequence {aₙ}ₙ₌₁∞ are given.

b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence).

{1, 2, 4, 8, 16, ......}

27–34. Working with sequences Several terms of a sequence {aₙ}ₙ₌₁∞ are given.

c. Find an explicit formula for the nth term of the sequence.

{1, 2, 4, 8, 16, ......}

27–34. Working with sequences Several terms of a sequence {aₙ}ₙ₌₁∞ are given.

b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence).

{1, 3, 9, 27, 81, ......}