Register to reply

Question on the Cauchy Condensation test

by Simfish
Tags: cauchy, condensation, test
Share this thread:
Simfish
#1
Feb6-08, 10:29 PM
PF Gold
Simfish's Avatar
P: 828
So it's here
http://en.wikipedia.org/wiki/Cauchy_condensation_test

My question is this: is the value of the base 2 in 2^k an arbitrary value? Or is there something special about 2? Can we just use something like e^k instead?
Phys.Org News Partner Science news on Phys.org
What lit up the universe?
Sheepdogs use just two simple rules to round up large herds of sheep
Animals first flex their muscles
HallsofIvy
#2
Feb7-08, 06:12 AM
Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,533
The 2 is not important, it's the exponential that is crucial.
cliowa
#3
Feb7-08, 06:13 AM
P: 191
Quote Quote by Simfishy View Post
So it's here
http://en.wikipedia.org/wiki/Cauchy_condensation_test

My question is this: is the value of the base 2 in 2^k an arbitrary value? Or is there something special about 2? Can we just use something like e^k instead?
Notice that the function f in [itex]\sum_{n=0}^{\infty} 2^{n}f(2^{n})[/itex] need not be well-defined for all arguments in the real numbers, but only for the natural numbers (including the zero).
What you have is a positive monotone decreasing sequence (which is denoted here by [itex]f(n)[/itex] but could just as well be written as [itex]a_n[/itex]). Now consider the sum (infinite series): the idea now is to form summation blocks with length [itex]2^n[/itex] and find a useful estimate for each block. Every block incorporates [itex]2^n[/itex] summands and the biggest summand in each block is the first one, as the sequence is monotone decreasing.

Try writing this down and see what you end up with. If you can, check the converse.


Register to reply

Related Discussions
Cauchy-Schwarz Inequality Proof Question Calculus & Beyond Homework 2
Cauchy Condensation Proof Calculus 6
Simple/trivial question on cauchy-riemann equations Calculus 2
Cauchy Riemann Conditions Question Calculus 1
Question about Condensation of Space General Physics 0