Limits of Sequences Homework: Proving Limit of a_n/n = 0

  • Thread starter Thread starter NickMusicMan
  • Start date Start date
  • Tags Tags
    Limits Sequences
NickMusicMan
Messages
7
Reaction score
0

Homework Statement



For a sequence a_n:

If lim (a_n) =2, use the definition of a limit to show that lim (a_n / n) = 0

all limits are as n goes to infinity

The Attempt at a Solution



I know that I need to show:

Give any \epsilon>0 there is some M so that

if n>M then |a_n / n| < \epsilonBut I can't Figure out how to do that.

Any help would be much appreciated! I have an exam in a few days and I came across this as a practice problem.

-NN
 
Last edited:
Physics news on Phys.org
by the way: I totally understand intuitively why this is the case, I just figure out how to express it formally.

I know that as n approaches infinity, the fraction approaches (a fixed number)/(infinity) , which means it approaches 0. How can I write this using the formal definition?
 
You know the limit of an, so you know there exists an N such that for n>N, |an-2|<ε.

Can you tweak that to get something that'll bound |an/n| from above? Or maybe use the triangle inequality?
 
I have tried using the triangle inequality to do so, but i haven't figured anything out yet.
 
NickMusicMan said:
I have tried using the triangle inequality to do so, but i haven't figured anything out yet.

Pick a value of epsilon. Then just divide the formal definition of limit by n.
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
Back
Top