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limits

if the limit of sequence a is infinity, and the limit of sequence b is c, how do i show the limit of sequence ab is infinity? also c>0

(edited 8 years ago)

Reply 1

joostan

Original post by asdfyolo

if the limit of sequence a is infinity, and the limit of sequence b is c, how do i show the limit of sequence ab is infinity? also c>0

There are a few different ways to do it I should imagine, some brief thoughts on the subject - you'll want to be more rigorous than I have been.
A brief sketch is that

(b_n)

converges so it is eventually close to

c

.
Since this is the case, and you know that

(a_n)

is divergent, you can see that, for large

n

the sequence

(a_nb_n) \sim c(a_n)

and so must diverge. If you can express that more carefully, then you are done.

Bigger hint in the spoiler - try to prove it yourself first. :smile:

[spoiler]Eventually you have

b_n>c-\epsilon

Spoiler

(edited 8 years ago)

Reply 2

BlueSam3

An alternative approach: You may already have done some of these steps, in which case you can skip them.

1) Prove that if

c > 0

(a_n)

a sequence diverging to

+\infty

, then

(ca_n)\to 0

.
2) Prove that if

(a_n)

is a sequence diverging to infinity, and

(b_n)

is a sequence such that

b_n > a_n

for all sufficiently large

n

, then

(b_n)\to 0

.
3) Since

(b_n)\to c

, there is in particular some

N

in the naturals such that for all

n > N

b_n > c/2 > 0

. Use this, and the above, to prove the result.

Reply 3

DFranklin

[QUOTE="joostan;60495845"]There are a few different ways to do it I should imagine, some brief thoughts on the subject - you'll want to be more rigorous than I have been.
A brief sketch is that