This discussion board is a place to discuss our Daily Challenges and the math and science
related to those challenges. Explanations are more than just a solution — they should
explain the steps and thinking strategies that you used to obtain the solution. Comments
should further the discussion of math and science.
When posting on Brilliant:
Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.
Markdown
Appears as
*italics* or _italics_
italics
**bold** or __bold__
bold
- bulleted - list
bulleted
list
1. numbered 2. list
numbered
list
Note: you must add a full line of space before and after lists for them to show up correctly
The condition that you have given is for asymptotic convergence. Without an additional condition of ∣xn+1−a∣≤k∣xn−a∣ the limiting condition cannot be proved.
With my knowledge of dynamics, there seems to be no solution without apriori assumption that limn→∞xn exists and is finite.
@Janardhanan Sivaramakrishnan
–
That's right, without any other conditions the only inequation we need to prove That a sequence converges to some value is the one You provided, and that's how we usually solve for the limits. Furthermore, we use This frequently when we have a sequence un satisfying:
un+1=f(un)
For some continuous and differentiable function f.
Easy Math Editor
This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.
When posting on Brilliant:
*italics*or_italics_**bold**or__bold__paragraph 1
paragraph 2
[example link](https://brilliant.org)> This is a quote# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"\(...\)or\[...\]to ensure proper formatting.2 \times 32^{34}a_{i-1}\frac{2}{3}\sqrt{2}\sum_{i=1}^3\sin \theta\boxed{123}Comments
∣xn+1−xn∣≤k∣xn−xn−1∣,∣xn−xn−1∣≤k∣xn−1−xn−2∣ Combining, ∣xn+1−xn∣≤k2∣xn−1−xn−2∣
This would imply that ∣xn+1−xn∣≤kn∣x1−x0∣.
Since, k∈[0,1], kn→0 as n→∞.
Hence, ∣xn+1−xn∣→0 as n tends to infinity.
The series is thus convergent.
Log in to reply
I did the same proof but my teacher told me it must be shown that lim(x_n)=a
Log in to reply
The condition that you have given is for asymptotic convergence. Without an additional condition of ∣xn+1−a∣≤k∣xn−a∣ the limiting condition cannot be proved.
With my knowledge of dynamics, there seems to be no solution without apriori assumption that limn→∞xn exists and is finite.
Log in to reply
That's right, without any other conditions the only inequation we need to prove That a sequence converges to some value is the one You provided, and that's how we usually solve for the limits. Furthermore, we use This frequently when we have a sequence un satisfying: un+1=f(un) For some continuous and differentiable function f.