Thread: Cauchy/convergent sequence

Question: Let {a_n} be a sequence of rational numbers defined as follows:

a_1 = 1
a_(n+1)=a_n + 1/(3^n)

for all n > 1.

1) Show that {a_n} is a Cauchy sequence and hence convergent
2) Find it's limit.


So.. I am a bit lost for this one. I tried using induction to bound a_n by 1 and 2, but I couldn't bound it above. I figured if I bound it above, and prove that it's a non-decreasing function, then I could prove that it converges, and thus that it is Cauchy...

Any tips for me to get started on this? Thanks!

Originally Posted by limddavid

Question: Let {a_n} be a sequence of rational numbers defined as follows:

a_1 = 1
a_(n+1)=a_n + 1/(3^n)

for all n > 1.

1) Show that {a_n} is a Cauchy sequence and hence convergent
2) Find it's limit.


So.. I am a bit lost for this one. I tried using induction to bound a_n by 1 and 2, but I couldn't bound it above. I figured if I bound it above, and prove that it's a non-decreasing function, then I could prove that it converges, and thus that it is Cauchy...

Any tips for me to get started on this? Thanks!

The difference equation can be written as...

(1)

... and its solution is...

(2)

From (1) and (2) You derive that is...

(3)

Kind regards

riight! sum of geometric series. I'm not 1000% sure I can use this in my class right now, but it's worth a try. thank you!