1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Proving ln(x) using infinite series

  1. Mar 31, 2008 #1
    1. The problem statement, all variables and given/known data

    Well we are given a series of steps done with the number "x" and in the end the end value is ln(x). Basically we are asked to prove why it isnt a coincedience

    2. Relevant equations

    I put the steps into an equation, but i cant prove it.

    ln(x) =[tex]^{lim }_{n->inf}[/tex] [tex](x^\frac{1} {2^n} -1)*2^n[/tex]

    3. The attempt at a solution

    Well plugging in gives me inf times 0 so i thought of solving it using Hopital's rule but i cant get to a form where it is 0/0. I tried factoring, rationalizing but i couldnt get anywhere. I would appreciate if you guys can help me do this little bit.

    and I would also appreciate if anyone can show me how to come up with a sequence for this. Second part of the question askes me to come up with my own limit of a sequence to get ln(x). I just flipped changed the sine of - inside the brackets to make it + and added the minus sign to the -2n at the end, but thats like the same thing. So if you guys can give me some ideas it would be great!! :)

    Thank you
    Last edited: Mar 31, 2008
  2. jcsd
  3. Mar 31, 2008 #2


    User Avatar
    Science Advisor

    Since n only occurs in 2n, you can simplify by letting a= 2n. Then the problem becomes showing that
    [tex]ln(x)= \lim_{a\rightarrow \infty}(x^(1/a)-1)(a)[/tex]

    Now I would be inclined to "reverse" the function: If y= (x1/a-1)(a), then x= (y/a+ 1)a. Do you recognize the limit of that as a common formula for ey? And if x= ey, then y= ln(x).
  4. Mar 31, 2008 #3
    tanks a bunch ill get back on you on the reverse function let me work it out. Im eating right now :) But the limit makes sense (argg why didn't i think of that)
  5. Mar 31, 2008 #4
    hmm i have a question about this, x= (y/a+ 1)a.

    Wouldn't x=1 when we take the limit as a>>inf?

    Thanks for your help

    never mind i see how x wouldn't equal to 1 because y has some "a" terms in it so we don't know the ratio. But i dont see how x = e^y. Is the x formula like an equation for the e function?
    Last edited: Mar 31, 2008
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook