Sequence and Series Problems

Hello everyone

I'm having trouble with these three questions and would like some help

the questions are :

(a) Can you give an example of a sequence which is increasing,

bounded from above and divergent? Justify your answer.

(b) Can you give an example of a convergent sequence {an} such that

sum ak , k=0 to infinty is divergent? Justify your answer.

(c) Can you give an example of a divergent sequence {an} such that

sum ak , k=0 to infinty is convergent? Justify your answer.

for the first question i thought of (-1)^n but its bounded from above and below and can't prove that it is increasing

for second question i think 1/n is the answer , as the sequence converges to 0 but the series diverges because it's a harmonic series

for the third question i have no idea at all

Thanks in advance !

Re: Sequence and Series Problems

b) The harmonic series is divergent while its terms converge to 0.

c) No, in order for a series to be convergent, the terms HAVE to converge to 0.

Re: Sequence and Series Problems

Quote:

Originally Posted by

**Lawati9** the questions are :

(a) Can you give an example of a sequence which is increasing,

bounded from above and divergent? Justify your answer.

(b) Can you give an example of a convergent sequence {an} such that

sum ak , k=0 to infinty is divergent? Justify your answer.

(c) Can you give an example of a divergent sequence {an} such that

sum ak , k=0 to infinty is convergent? Justify your answer.

for the first question i thought of (-1)^n but its bounded from above and below and can't prove that it is increasing

for second question i think 1/n is the answer , as the sequence converges to 0 but the series diverges because it's a harmonic series

for the third question i have no idea at all

a) There is a theorem: *A bounded monotone sequence converges.*

b) You are correct.

c) There is a theorem: *$\displaystyle \sum\limits_{n = 1}^\infty {{a_n}}$ converges only if $\displaystyle (a_n)\to 0~.$*