I am a Calc 2 tutor and I wanna give my tutees an obnoxiously hard problem to work on. We are doing Infinite series now...

I am a Calc 2 tutor and I wanna give my tutees an obnoxiously hard problem to work on. We are doing Infinite series now. Give me what you got, /sci.

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math.stackexchange.com/questions/709483/pointwise-converging-subsequence-on-countable-set
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Prove that any sequence of functions f_n : R -> [0, 1] has a pointwise convergent subsequence.

slyly introduce them to laplace transforms.

provve 1 + 1 = 2 using principles of FOL

ask them to sum 1+2+3+4+...

let n be some number in the sequence.

limit as n goes to infinity is infinity.

sum of n terms is infinity

*leans into mic*

WRONG!

it's -1/12, obviously

\sum_{k=1}^{n}\left ( \frac{1}{k}\left ( \binom{n}{k}+1 \right ) \right )=\sum_{k=1}^{n}\frac{2^k}{k}

A clever Calc I student could do this

not terribly difficult but a fun one:

e^(x+e^x))

Ask them to prove the sum of the natural numbers converges to -1/12

Let [math]a_0,a_1,...,a_n\in \mathbb{C}[/math] be a collection of [math]n+1[/math] complex numbers with [math]n \geq 1[/math] and [math]a_n \neq 0[/math]. Let [math]p(z)=a_0+a_1z+...+a_nz^n[/math]. Prove the existence of a point [math]z_0 \in \mathbb{C}[/math] such that [math]p(z_0)=0[/math].

sum of the reciprocal of squares, no problem.

or better yet, sum of reciprocal of cubes

Nah this too easy. Just follows from Bolzano theorem.

How about:
Ask them to integrate simple functions like sin(x) over null sets?

legitimately a good and constructive idea

Proof:

- The field C, is defined as the algebraic closure of R. Thus it is naturally closed and the theorem is trivial. QED

Bolzano says that for any individual point x there is a subsequence so that f_m(x) converges, but not that you can do this for all x simultaneously. It's a special case of Tychonoff's theorem, which isn't that hard but considering it uses AC it's a troll problem for Calc 2 students.

Prove P=NP.

Enumerate your language and denote by [math] h(n,m) \in \{0, 1\} [/math] whether the n'th Turing machine halts before m steps (1 for halts, 0 for doesn't halt). That's a function
[math] h : {\mathbb N} \time s{\mathbb N} \to \{0,1\} [/math]

Let's define the sequence
[math]s_k := \sum_{n+m=k} \frac{1}{2^n} h(n,m) [/math]

Compute
[math] s := \lim_{k\to \infty} s_k [/math]

Why the fuck you do this

Just keep with more elaborated but not that hard questions, you cunt

Fuck

math.stackexchange.com/questions/709483/pointwise-converging-subsequence-on-countable-set

Looking at whole proof is tough kek

That's only for a countable domain though, not all of R. There probably is a more direct way of doing this than proving Tychonoff but I'm pretty sure you need some form of choice.

Circular logic.

He didn't define C. So I chose a definition making the problem trivial.

This is Formal Languages and Automata Theory you cunt