Question 9(3 points)

Consider the following statements:

(i)If

n=1

a

n

{"version":"1.1","math":"\(\sum_{n=1}^\infty a_n\)"}converges, then

n=1

|a

n

|

{"version":"1.1","math":"\(\sum_{n=1}^\infty |a_n|\)"}converges.

(ii)If

n=1

a

n

{"version":"1.1","math":"\(\sum_{n=1}^\infty a_n\)"}is an alternating series andlim

n

a

n

{"version":"1.1","math":"\(\lim_{n\to\infty}a_n\)"}= 0, then

n=1

a

n

{"version":"1.1","math":"\(\sum_{n=1}^\infty a_n\)"}converges.

(iii)Iflim

n

|a

n+1

a

n

|=L

{"version":"1.1","math":"\(\lim_{n\to\infty}| \frac{a_{n+1}{a_n} |=L\)"}exists andL> 1, then

n=1

a

n

{"version":"1.1","math":"\(\sum_{n=1}^\infty a_n\)"}converges.

(iv)Iflim

n

a

n

0

{"version":"1.1","math":"\(\lim_{n\to\infty}a_n e0\)"}, then

n=1

a

n

{"version":"1.1","math":"\(\sum_{n=1}^\infty a_n\)"}diverges.

Which one(s) of the these statements is/are true?

Question 9 options:

a)

(i) and (iv) only.

b)

All of them.

c)

(i), (ii), and (iv), only.

d)

(iv) only.

e)

None of them.

f)

(i) and (ii) only.

Question 9 (3 points) Consider the following statements: (i) If Z'n=1 an converges, then En=1 lanl converges. (ii) If En=1 an is an alternating series and limn too an= 0, then En= an converges. (iii) If limn co | an anti | = L exists and L > 1, then Z'n=1 an converges. (iv) If limn too an # 0, then En=1 an diverges. Which one(s) of the these statements is/are true? O a) (i) and (iv) only. ( b) All of them. c) (1), (ii), and (iv), only. O d) (iv) only. ( e) None of them. Of) (i) and (ii) only