Which of the following sequences diverges?

Question

Which of the following sequences diverges?

 

a) an=1/n

b) an=((-1)^(n+1))/n

c) an=(2^n)/(e^n)

d) an=(n^2)/(e^n)

e) an=n/ln(n)

 

Answer: E

 

n after an is lower case.

 

Please show all your work. Thanks.

Answer 1

Which of the following sequences diverge?

a) an=1/n

n → ∞ ==> 1/n  → 0 ==> convergent

b) an=((-1)^(n+1))/n=((-1)^(n+1))*(1/n)

(-1)^(n+1) = {–1 if n even, +1 if n odd}

an = ±1/n  → 0 as n  → ∞ ==> convergent

c) an=(2^n)/(e^n)= (2/e)^n

2/e < 1, an  → 0 as n  → ∞ ==> convergent

d) an=(n^2)/(e^n)

n → ∞ ==> (n^2)/(e^n) → ∞/∞; use L’Hospital twice.

(n^2)/(e^n) → 2n/e^n → 2/e^n → 0 ==> convergent

e) an=n/ln(n); use L’Hospital

n/ln(n) → 1/(1/x) = x → ∞ ==> divergent.

Answer: E √