I have, since posting the question, tried to find an upper bound and lower bound to the sequence, but I have TOTALLY failed when calculating the number of terms: ##2^{n+1} - 2^n = 2^n(2^n - 1)##, instead of ##2^{n+1} - 2^n = 2^n(2 - 1) = 2^n## confusing myself entirely and giving up trying to...
Homework Statement
Prove that ##\forall n \in \mathbb{N}##
$$\frac{n}{2} < 1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{2^n - 1} \leq n \text{ .}$$
Homework Equations
Peano axioms and field axioms for real numbers.
The Attempt at a Solution
Okay so my first assumption was that this part...
So I am about to switch to mathematics, as I absolutely adore pure mathematics as well as applied mathematics and I cannot stand learning mathematics without going deep into rigor of it for now(I am aware however that at times one just has to quickly learn sth to proceed with a project). But in...
Hmm so I was right to hypothesise that we cannot prove this theorem with naturals starting at 1 as 1 fails the definitions of oddness and evenness, except of course we define oddness of n as n+1=2k for some k, then 1 satsfies it perfectly. I did not even think about this solution!
Thank you :)
So I was just writing a proof that every natural number is either even or odd. I went in two directions and both require that 1 is odd, in fact I think that 1 must always be odd for every such proof as the nature of naturals is inductive from 1.
I am using the version where 1 is the smallest...
Thank you very much! I do fully agree on the importance of the foundations of oneself's knowledge. I do have a lot of experience in programming so I do know how important this is. That's why I think option 2 will be the best. I will work through books(I LOVEEE Spivak's Calculus, it made me fall...
I am a physics freshman in Europe where we have 3 year undergrad programmes. I am just over the first year of Multiple Sclerosis and just when the school started in october last year, things went very very bad. I was super ready for UNI and I started off very well with awesome routines, but I...
Our system is exactly the same, after all, Slovenia's Bologna System was designed after your German one. I have certainly not "decided" in which direction I want to go yet, since as you said it, I don't have nearly enough knowledge to know. For now though, I do prefer theory and mathematical...
I thought so, I admit I did not do my research regarding this and was rather naive. Well you needn't have been toxic though(as you are implying I will fail bad by saying I will study even up to 8 years...), it will take me 3 years, hopefully. I might have missed your point though.
You are absolutely right, not a lot of rigor is useful in physics, however how am I supposed to understand and properly apply topological ideas to physics if I can't even understand the construction of real numbers for example. I want to trully understand how math works so I am rather happy for...
Ah ok, I think I understand, it is true we had basic physics, however without any calculus, so practically we learned nothing. We did learn very basic calculus by the end of the last year of high school though. So far it seems Kleppner should be awesome for mechanics, but I do have two good...
Well, its same as pure math major except they aim at "all" of math and we only analysis and some topology it seems. As I see it, its is extremely useful for a theoretical physicist to be excelent at pure math and should know as much math as possible; hence we are learning all of that. We will...
Just so that we are on the same page, our curriculum is only 3 years, so I suppose things will be different here, at least more condensed learning I suppose. Morever, many many students got to grad in US and they all say they got very solid foundations from the UNI here. What do you guys think?