• Support PF! Buy your school textbooks, materials and every day products Here!

Exponential function

  • #1
125
0

Homework Statement


For the exponential function [itex]exp(z) = \sum^{\infty}_{n=0} \frac{z^n}{n!}[/itex], we know that [itex]exp(z+w)=exp(z).exp(w)[/itex], and that, if [itex]x \geq 0[/itex], then [itex]exp(x) \geq 1+x[/itex]. Use these facts to prove that, if [itex]1 \leq n \leq m[/itex], then

[tex]|\prod^n_{j=1} (1+a_j) - \prod^m_{j=1} (1+a_j)| \leq exp(\sum^n_{j=1} |a_j|) . (exp(\sum^m_{j=n+1} |a_j|)-1)[/tex]

Homework Equations





The Attempt at a Solution



I simplified to:

[tex]|a_{n+1}+a_{n+2}+...+a_m+a_{n+1}a_{n+2}+...+a_{n+1}a_{n+2}...a_m| \leq \sum^{\infty}_{n=0} \frac{(\sum^m_{j=1} |a_j|)^n}{n!} - \sum^{\infty}_{n=0} \frac{(\sum^n_{j=1} |a_j|)^n}{n!}[/tex]

from here I had the idea that I could break n=0 and n=1 out of the exponentials since n=0 doesnt yield anything and n=1 yields some individual terms i could subtract from the left side. I am thinking this is the wrong approach though
 
Last edited:

Answers and Replies

Related Threads on Exponential function

  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
4
Views
7K
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
5
Views
2K
  • Last Post
Replies
1
Views
818
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
2
Views
861
  • Last Post
Replies
3
Views
570
Top