Check all of the following that are true for the series

Question

Check all of the following that are true for the series from n=1 to ∞ ∑ (5-cosn)/n.

A. This series converges

B. This series diverges

C. The integral test can be used to determine convergence of this series.

D. The comparison test can be used to determine convergence of this series.

E. The limit comparison test can be used to determine convergence of this series.

F. The ratio test can be used to determine convergence of this series.

G. The alternating series test can be used to determine convergence of this series.

Please help

Answer 1

For all natural numbers n,

-1 ≤ cos n ≤ 1

⇒ -1 ≤ -cos n ≤ 1

⇒ -1 + 5 ≤ -cos n + 5 ≤ 1 + 5

⇒ 4 ≤ 5 - cos n ≤ 6

⇒ 4 / n ≤ (5 - cos n) / n

⇒ ∑_n 4 / n ≤ ∑_n (5 - cos n) / n

∑_n 4 / n = 4 * ∑_n 1 / n

We know that the Harmonic series diverges, therefore we immediately know that ∑_n 4 / n diverges as well.

By a Comparison Test, we can conclude that ∑_n (5 - cos n) / n diverges.

So B and D are true.

A is False.

The rest is a little tricky...

So instead of the alphabetical order, I am putting them in order from easiest ones to explain to hardest ones to explain.

G is false.

The series that we are looking at is clearly not an alternating series.

C is false.

The integral ∫ [(cos x) / x] dx does not have a closed form antiderivative.

So an integral test would not work.

E is false.

This is due to the nature of cosine: as n → ∞, cos n is bounded but does not converge to any particular value. So we will not get a nice finite nonzero limit involving cosine.

F is false.

The ratio test would yield an inconclusive result.

We would get (5 - cos (n + 1)) / (5 - cos (n)) and n / (n + 1).

The limit of n / (n + 1) as n → ∞ is 1 which is inconclusive in a ratio test.

(5 - cos (n + 1)) / (5 - cos (n)) is bounded but does not converge to any particular value due to the nature of cosine.

So the ratio test would not yield a proper limit.

Answer 2

4/n ≤ (5 - cosn)/n

But left series diverges as harmonic series. So, given series divergent (comparison test).