cauchy sequence

sara_87

1. The problem statement, all variables and given/known data

Show that the following sequence converges:

xn= (sin(1)/2) + (sin(2)/2^2) + (sin(3)/2^3) +...+ (sin(n)/2^n)
2. Relevant equations



3. The attempt at a solution
To show that it converges, i want to show that it is a cauchy sequence (since all cauchy sequences converge).
I know that xn is cauchy if abs(xn-xm)< E for all E>0.
and the above sequence can be written as:
[tex]\sum(sink/2^k[/tex]

But i dont know how to proceed??
Any help would be very much appreciated.

dirk_mec1

Is that a fact? Check really if that's true...

As for your sequence, your sequence is (monotonically) increasing and bounded thus convergent. Can you show that it is true?

Dick

It's not monotonic. sin(k) can be negative. And why are you questioning whether cauchy sequences converge?? The question is clearly about the real numbers, and they are complete. All cauchy sequences DO converge. sara_87, to show it's cauchy just notice sin(k)/2^k<=1/2^k. You can bound abs(xn-xm) by the sum of a geometric series.

dirk_mec1

Yes, you're right I made two mistakes.

Dick uses [tex] sin(x) \leq 1[/tex]

Dick

Yep. And I should have written |sin(k)/2^k|<=1/2^k.

sara_87

ok, so |sin(k)/2^k|<=1/2^k

and the formula for geometric series is:
a(1-r^n)/1-r

i dont know how to apply this to sin(k)/2^k

dirk_mec1

Look, you want to have [tex]|a_n - a_m|< \varepsilon [/tex].

Now you know you can make an upper bound via a geometric series, okay?

sara_87

Yes, i understand this bit and i know that in this case, we have:
abs(sin(k)/2^k) < 1/2^k

but is this it?? surely not. I still havent proved that it is cauchy.

Mark44

The problem as stated asked that you prove that the sequence was convergent. You brought up the part about Cauchy sequences. If you can show that the sequence converges (see Dick's and Dirk's posts), you're done.

sara_87

Yes, but i 'want' to show that it is cauchy (part of the question). I know that:
abs(xn-xm)< E
and
abs(sin(k)/2^k) < 1/2^k
but how do i relate these two?

Mark44

Using your definition for x_n at the beginning of this thread, what is x_n - x_m?

sara_87

(sin(1)/2) + (sin(2)/2^2) + (sin(3)/2^3) +...+ (sin(n)/2^n) - xm

Dick

Let's assume n>m. So xn-xm=sin(m+1)/2^(m+1)+sin(m+2)/2^(m+2)+...sin(n)/2^n. |xn-xm|<=1/2^(m+1)+...+1/2^n. Apply your geometric series thing to that.

sara_87

ok, so i get:
abs(xn-xm)<=1/2^m for all m>N ;
but this must mean that 1/2^m is less than E but how do we know that m is greater than E (for this to be true)?

Dick

You are given e. You PICK an N large enough that 1/2^N<e. That means |xn-xm|<e for all n,m>N, right? That's Cauchy.

sara_87

but however large N is, the 1/2^N will always be postive.
so, just as an example, what kind of value can N be?

dirk_mec1

N will depend on your epsilon chosen. In your case you should use the geometric series to find your N.