Exponential function proof problem

[rustyjoker]

Homework Statement

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]

 

[Ray Vickson]

Homework Statement

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]

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

 

[rustyjoker]

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]