Converging and Diverging Sequences WORK SHOWN... NEED HELP!

[djdavis2k]

Give an example of two sequences {a_n} and {b_n} such that:

a.) a_n and b_n are divergent, but (a_n+ b_n) is convergent

b.) a_n is convergent , b_n is divergent ,and (a_n)(b_n) is divergent

c.) a_n is divergent, but |a_n| is convergent

my work:

for c.) if a_n= (-1)^n the sequence is symbolic of a harmonic alternating series, therefore the sequence is divergent as it alternates between the values of 1 and -1 as n approaches infinity.. so it is diverging and a limit does not exist.. however if the absolute value of (a_n) is taken the value converges to +1 as n approaches 1... i believe this is an okay example.. please mention any assumption I've done that may be wrong here

for a.) is this a correct example?.... if a_n is equal to )1/n )and b_n is equal to (n) therefore b_n is divergent since it doesn't equal 0 as n approaches inf. for the limit since it approaches n, however 1/n approaches 0 as n approaches inf. so it is convergent... however the a_n*b_n is divergent since it's product is n which is divergent... i think i made a mistake in my assumption here...


for b.) i need a lot of help here

thanks for your help in this calculus form.. you have helped a great deal and i've learned a lot from this helpful forum.. can you please help me with these problems... and correct the mistakes i've made

 

[Deleted member 4993]

Give an example of two sequences {a_n} and {b_n} such that:

a.) a_n and b_n are divergent, but (a_n+ b_n) is convergent

b.) a_n is convergent , b_n is divergent ,and (a_n)(b_n) is divergent

c.) a_n is divergent, but |a_n| is convergent

my work:

for c.) if a_n= (-1)^n the sequence is symbolic of a harmonic alternating series, therefore the sequence is divergent as it alternates between the values of 1 and -1 as n approaches infinity.. so it is diverging and a limit does not exist.. however if the absolute value of (a_n) is taken the value converges to +1 as n approaches 1... i believe this is an okay example.. please mention any assumption I've done that may be wrong here

for a.) is this a correct example?.... if a_n is equal to )1/n )and b_n is equal to (n) therefore b_n is divergent since it doesn't equal 0 as n approaches inf. for the limit since it approaches n, however 1/n approaches 0 as n approaches inf. so it is convergent... however the a_n*b_n is divergent since it's product is n which is divergent... i think i made a mistake in my assumption here...


for b.) i need a lot of help here

thanks for your help in this calculus form.. you have helped a great deal and i've learned a lot from this helpful forum.. can you please help me with these problems... and correct the mistakes i've made

For a) choose a telescoping series like:

\(\displaystyle \sum_{n=1}^{\infty}[\frac{1}{n} \, - \, \frac{1}{n+1}]\)

for b)

as I read it assume:

\(\displaystyle a_n \, = \, \frac{1}{n^2}\)

and

\(\displaystyle b_n \, = \, n^3\)