24 Feb '04 05:53

In the process of trying to prove the convergency of the sum of a sequence--or, more spacifically, the divergency of it--I attempted to change the function that I was working with into something I was familiar with, but in doing so I ended up creating a proof which seems very wrong--mainly because the proof suggests that 0=1, which clearly isn't true. I was hopeing that someone could prove my proof wrong because it is bugging me, and I have no clue why it is wrong.

And since it is wrong (or so I assume it is) can someone actually prove by other means why the particular function is divergent.

Here goes:

I had to prove why the following function is divergent (has no definate value):

Sum (x=1 to infinity) sin(1/x)

I know that this graph looks like a decrasing exponential function from x=1/pi to infinity (for those who don't know, an exponential function is a function that can be written as x^n, where n is a constant and x is a variable; a decreasing geometric series has r: r<0). I know how to prove whether an exponential function is convergent or divergent so I attempted to change the function into one.

n is an arbitrary variable:

sin(1/x)=1/x^n

ln(sin(1/x))=n(ln(1/x))

n=ln(sin(1/x))/ln(1/x)

I believe that if I could determin what the value of n was as x approached infinity, it would help me to determin whether the function was convergent or divergent, so I did just that.

lim (x->infinity) n=?

lim (x->infinity) ln(sin(1/x))/ln(1/x)

this value becomes undefined/infinity, therefore I needed to use the h' pitol (I have no clue what the guy's name is or how to spell it) rule to find the actual limit.

lim (x->infinity) (-1/x^2(cos(1/x)/sin(1/x)))/(-1/x^2/(1/x))

lim (x->infinity) tan(1/x)/x

this becomes 0/infinity, which equals 0; therefore:

lim (x->infinity) n=0

According to this:

lim (x->infinity) 1/x^n = 1/(infinity)^0 = 1

yet:

lim (x->infinity) sin(1/x) = 0

If the two functions are equal than how can the above limits be different? They must be equal because I set them equal to each other and solved for n, yet they are not. Why?

Please explain to me why the above proof is wrong, and please also explain why the summation of sin(1/x) from 1 to infinity is divergent. I also used a rule in my proof, the H' pitol rule (or something like that), what is the actual name of that rule?

P.S. I am a AP Calculus BC student, so if you are able to explain this, please do so in terms that I can understand.

Thanks

Econundrum

And since it is wrong (or so I assume it is) can someone actually prove by other means why the particular function is divergent.

Here goes:

I had to prove why the following function is divergent (has no definate value):

Sum (x=1 to infinity) sin(1/x)

I know that this graph looks like a decrasing exponential function from x=1/pi to infinity (for those who don't know, an exponential function is a function that can be written as x^n, where n is a constant and x is a variable; a decreasing geometric series has r: r<0). I know how to prove whether an exponential function is convergent or divergent so I attempted to change the function into one.

n is an arbitrary variable:

sin(1/x)=1/x^n

ln(sin(1/x))=n(ln(1/x))

n=ln(sin(1/x))/ln(1/x)

I believe that if I could determin what the value of n was as x approached infinity, it would help me to determin whether the function was convergent or divergent, so I did just that.

lim (x->infinity) n=?

lim (x->infinity) ln(sin(1/x))/ln(1/x)

this value becomes undefined/infinity, therefore I needed to use the h' pitol (I have no clue what the guy's name is or how to spell it) rule to find the actual limit.

lim (x->infinity) (-1/x^2(cos(1/x)/sin(1/x)))/(-1/x^2/(1/x))

lim (x->infinity) tan(1/x)/x

this becomes 0/infinity, which equals 0; therefore:

lim (x->infinity) n=0

According to this:

lim (x->infinity) 1/x^n = 1/(infinity)^0 = 1

yet:

lim (x->infinity) sin(1/x) = 0

If the two functions are equal than how can the above limits be different? They must be equal because I set them equal to each other and solved for n, yet they are not. Why?

Please explain to me why the above proof is wrong, and please also explain why the summation of sin(1/x) from 1 to infinity is divergent. I also used a rule in my proof, the H' pitol rule (or something like that), what is the actual name of that rule?

P.S. I am a AP Calculus BC student, so if you are able to explain this, please do so in terms that I can understand.

Thanks

Econundrum