Question

Show that this sequence is monotone or eventually monotone by using the Monotone Convergence Theorem. (Proof)

n/(3^n)

Answer 1

Let Xn = n/3^n

Xn+1 = (n+1)/3^(n+1)

Now

Xn+1 - Xn = (n+1)/3^(n+1) - n/3^n

= [(n+1)-3n]/3^(n+1)

= (1-2n)/3^(n+1)

< 0 (since X1 =(1/3), X2=(2/9)==>X1>X2)

==> Xn+1 - Xn <= 0

==>Xn+1<=Xn, for every n

==> (Xn) is a decreasing sequence of real numbers.

==>(Xn) is a Monotone Decreasing Sequence of real numbers.

We now that Monotone Decreasing Sequence has Bounded above.

Now we have to prove that (Xn) has Bounded below

==> lim n-->infinity (n/3^n)

If we apply limit that convert into (inf/inf) form. For that we have to use L - Hospital rule then

lim n-->inf. (dn/dn) /(d3^n/dn) = lim n-->inf (1/3log3^n)

=1/inf

=0

0<=Xn<1/3

The sequence (Xn) convergence to 0

==> (Xn) =n/3^n is a Bounded Monotone Convergence a sequence.

Hence proved