The Proof is Trivial!
Scroll to see replies
Spoiler
Oh effing eff - I meant abscissa - that'll teach me to use those fancy college kid words. I'll edit.
It is spoilered. . .?
Ignore me. I'm tired and heading off to bed. I look forward to your complete argument in the morning.
Using the standard notation here to save TeXing all this junk.
Spoiler
Minor typo, can't be negative - but lovely solution! This notation fuzzles it up enough for me not to understand it.
whupsy daisy -that was rather than , will edit.
It is a little confusing, but if you write it out it's simple enough, I was being lazy.
That looks fine, but I think that there's a much simpler argument:
Spoiler
Spoiler
It is a little smoother I guess, but I calculated the ordinates first.
Problem 572:*
Compute the exact value of:
Where this fraction continues in this recurring way ad infinitum.
Find an expression of this type (known as a continued fraction) for , and .
Problem 572:*
Yaay! Iterative stuff!
Where this fraction continues in this recurring way ad infinitum.
\displaystyle [br]\begin{equation*}x -13 = \frac{1}{2 + \frac{1}{26 + x-13}} \Rightarrow \frac{1}{x-13} = 2 + \frac{1}{x+13} \Rightarrow \frac{1}{x-13} - \frac{1}{x+13} = 2 \end{equation*}
So that where we pick the positive root for obvious reasons.
We have:
\displaystyle [br]\begin{equation*}2x = 1 + \sqrt{5} \Rightarrow (2x-1)^2 = 5 \Rightarrow 4x^2 - 4x -4 = 0 \iff x^2 - x - 1 = 0\end{equation*}
So:
\displaystyle[br]\begin{equation*} x^2 = 1 + x \Rightarrow x = 1 + \frac{1}{x} = 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \cdots}}}\end{equation*}
We have:
\displaystyle [br]\begin{equation*}\sqrt{2} = 1 + \sqrt{2} - 1 = 1 + \frac{1}{1 + \sqrt{2}} \Rightarrow \sqrt{2} = 1 + \frac{1}{2 + \frac{1}{2 + \frac{1}{2+ \cdots}}}
Determine whether
1/1^5 + 1/2^5 + 1/3^5...
is irrational.
Determine whether
1/1^5 + 1/2^5 + 1/3^5...
is irrational.
1^5 is irrational so the sum of an irrational number and some other numbers are irrational, hence the sum is irrational. QED.
^ me trying to learn analysis
Posted from TSR Mobile
Such proof. . .
Many brilliance. . .
Spoiler
\displaystyle [br]\begin{equation*}x -13 = \frac{1}{2 + \frac{1}{26 + x-13}} \Rightarrow \frac{1}{x-13} = 2 + \frac{1}{x+13} \Rightarrow \frac{1}{x-13} - \frac{1}{x+13} = 2 \end{equation*}
So that where we pick the positive root for obvious reasons.
We have:
\displaystyle [br]\begin{equation*}2x = 1 + \sqrt{5} \Rightarrow (2x-1)^2 = 5 \Rightarrow 4x^2 - 4x -4 = 0 \iff x^2 - x - 1 = 0\end{equation*}
So:
\displaystyle[br]\begin{equation*} x^2 = 1 + x \Rightarrow x = 1 + \frac{1}{x} = \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \cdots}}}\end{equation*}
We have:
\displaystyle [br]\begin{equation*}\sqrt{2} = 1 + \sqrt{2} - 1 = 1 + \frac{1}{1 + \sqrt{2}} \Rightarrow \sqrt{2} = \frac{1}{2 + \frac{1}{2 + \frac{1}{2+ \cdots}}}
gah, just as I finished my solution I refresh the page and you've got there first! Atleast I got the same answers
Ah, sorry - that's always frustrating!
Having looked more closely I now notice you seem to have made a typo.
In both cases you're missing the preceding .
In both cases you're missing the preceding .
Urgh, thanks! Fixed. :-)
Lol don't be sorry, I'll have to be faster next time
And I'll have to make less mistakes.
Quick Reply
Related discussions
- Proof by induction - general question - why prove n=1 works?
- I am the biggest ****** in the world, please remind me ocassionally in case I forget
- Proof question
- BMO 1993, Round 1, Question 3
- Discrete maths questions
- A level maths proof Q
- How should I go about preparing for the STEP Maths exam?
- Range of a function
- PDEs
- AI in robotics PS?
- The hard grade 9 maths questions thread 2023
- CV going into Uni
- How to practice GCSE OCR Computer Science J277 Paper 2?
- Procrastination
- English language paper 2 exam !!
- OCR a level exam question
- Demoivres theorem quick application
- Edexcel AS Level Further Maths 2022
- Is there a rule of thumb of when to use integration by parts or substitution?
- How would the mass at B work in this moments question?
Latest
Trending
Last reply 2 days ago
Did Cambridge maths students find maths and further maths a level very easy?Last reply 2 weeks ago
Edexcel A Level Mathematics Paper 2 unofficial mark scheme correct me if wrong71
Trending
Last reply 2 days ago
Did Cambridge maths students find maths and further maths a level very easy?Last reply 2 weeks ago
Edexcel A Level Mathematics Paper 2 unofficial mark scheme correct me if wrong71
