Exponential function proof problem

[rustyjoker]

Homework Statement

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

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

x_{k1}≥...≥x_{k_{nk}} and x_{k1}+...+x_{k_{nk}}=x.
Proof: if lim_{k→}∞ x_{k1}=0 then lim_{k→}∞<br /> <br /> (1+x_{k1})·...·(1+x_{k_{nk}})=exp(x)=e^{x}

 

[Ray Vickson]

Homework Statement

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

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

x_{k1}≥...≥x_{k_{nk}} and x_{k1}+...+x_{k_{nk}}=x.
Proof: if lim_{k→}∞ x_{k1}=0 then lim_{k→}∞<br /> <br /> (1+x_{k1})·...·(1+x_{k_{nk}})=exp(x)=e^{x}

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

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 lim_{k→}∞ n_{k}=∞, then <br /> <br /> lim_{k→}∞ x_{k1}=...=lim_{k→}∞ x_{k_{nk}}=x/n_{k}= 0. So we'd have
lim_{k→∞} (1+x/n_{k})^{n_{k}} = e^{x}