Question

please help with this question, provide with full solution. ​

Answer 1

Step-by-step explanation:

1. Calcdate

(ln n)

lim

Solution: Use L'Hospital's rule (twice):

2 ln r

(2lnz'

((ln

lim

lim

lim

lim

lim

2. Find

Solution:

4. 4'

3 5'

3. 5'

with a

and r

(The series is geom

3. Discuss the converyence of the series

(-1)t1

Vn

Solution: It converges conditionally. It's an alteruating series with b, 1//n.

n and thus b

We have (i) The sequer

ce [bnJ2 is decreasing since Vn +1

- limq- 1/ - 0. Thus the

1/Vn +11/ b for all 2. (ii) li

series converges by the Alternating Series Test. But the series

diverges since it's a p serics and p 1.

(-1)+1

4. Use Comprison Tests to determine which one of the following series is diveryent.

T a p-series with p

-couverges by comparison with

Solution:

1.

(b)

, a p-series with p 2 >1.

converges by comparison with

+8

a p-series with p

ndiverges by limit comparison with

+ 100