1. Not finding help here? Sign up for a free 30min 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!

Exponential function proof problem

  1. Jan 29, 2012 #1
    1. The problem statement, all variables and given/known data

    Big problem with exponential function proof assignment, need some help.

    let
    x≥0 and for every k[itex]\in N[/itex] there is [itex]n_{k}[/itex][itex]\in N[/itex] and

    [itex]x_{k1}[/itex]≥...≥[itex]x_{k_{nk}}[/itex] and [itex]x_{k1}[/itex]+...+[itex]x_{k_{nk}}[/itex]=x.
    Proof: if [itex]lim_{k→}∞ x_{k1}[/itex]=0 then [itex]lim_{k→}∞

    [/itex] (1+[itex]x_{k1}[/itex])·...·(1+[itex]x_{k_{nk}}[/itex])=exp(x)=[itex]e^{x}[/itex]
     
  2. jcsd
  3. Jan 29, 2012 #2

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    There must be something wrong with the statement of hypotheses, because it allows me to take [itex]x_1 = x,\: x_2 = x_3 = \cdots = x_n = 0,[/itex] giving [itex] \lim_{n \rightarrow \infty} (1+x_1) \cdot (1+x_2) \cdots (1+x_n) = 1+x.[/itex]

    RGV
     
  4. Jan 29, 2012 #3
    Well, for an example if you think about the product
    (1+0,009)(1+0,008)⋅...⋅(1+0,001)=1,045879514 and
    exp(0,009+...+0,001)=1,046...
    I think the idea is to proof that first [itex]lim_{k→}∞[/itex] [itex]n_{k}=∞, then

    [/itex] [itex]lim_{k→}∞[/itex] [itex]x_{k1}[/itex]=...=[itex]lim_{k→}∞[/itex] [itex]x_{k_{nk}}[/itex]=x/[itex]n_{k}[/itex]= 0. So we'd have
    [itex]lim_{k→∞}[/itex] (1+x/[itex]n_{k}[/itex])[itex]^{n_{k}}[/itex] = [itex]e^{x}[/itex]
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook