- #1

fresh_42

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## Summary:

- This is an extra round of questions, because the ones in part I had been solved so fast. However, we still have some easy ones here. They are from calculus, abstract algebra, algebraic geometry, and topology. (Authors: Infrared - IR; fresh_42 - FR)

## Main Question or Discussion Point

**Questions**

1.(solved by @hilbert2 ) Let ##\sum_{k=1}^\infty a_k## be a given convergent series with ##|a_{k+1}|\leq |a_k|## for all ##k##. Assume we use a computer to sum its value until the partial sum is closer than ##\varepsilon## to the actual value of the series. Does it make sense to use ##|a_n|<\varepsilon## as a stopping criterion for the loop? Please justify your answer. (FR)

1.

**2.**(solved by @Antarres ) "Every absolutely convergent series converges." Now why is its proof so complicated, couldn't we just say:

Given an absolute convergent series ##\sum_{k=1}^{\infty} |a_k|## then we have for the sequence ##R_n :=\sum_{k=n+1}^\infty a_k##

$$

|R_n| = \left|\sum_{k=n+1}^\infty a_k\right| \leq \sum_{k+1}^\infty |a_k|

$$

with the remainder of a convergent series on the right, hence a null sequence. Thus ##R_n## is a null sequence, too, and the series is convergent. (FR)

**3.**(solved by @archaic ) Calculate the limit (## i ## being the imaginary unit): (FR)

$$

\lim_{n \to \infty} \operatorname{Arg}\left(\sum_{k=0}^n \dfrac{1}{k+ i }\right)

$$

**4.**Show that there is no odd (##>1##) dimensional real division algebra ##D##. (FR)

**5.**Let ##R:=\mathbb{Z}_{(5)}=\left\{\left.{\dfrac{a}{b}}\ \right\vert \ 5\nmid b\right\}## the ring of rational numbers which don't have a factor ##5## in their denominator, ##M\neq \{0\}## a finitely generated ##R-##module, and ##I:=\left\{\left.{\dfrac{a}{b}}\in R\ \right\vert \ 25\mid a\right\}\,.##

Prove that ##I## is an ideal contained in the Jacobson radical of ##R## and that ##IM\neq M\,.## The Jacobson radical ##J=J(R)## is defined as the intersection of all maximal ideals. (FR)

Edit: Definition of ideal ##I## corrected to make it one. The mistake didn't affect the central statement.

**6.**Calculate (FR)

$$

\lim_{n \to \infty}\dfrac{\sqrt{n\pi}}{2^{2n}}\cdot \binom{2n}{n}

$$

**a.)**(solved by @Fred Wright ) without using Stirling's formula.

**b.)**by using Stirling's formula, with accurate remainder terms, i.e. not simply ##\sim ##.

**7.**Find the minimal real number ##A## such that if ##f:[0,1]\to\mathbb{R}## is any ##C^1## function satisfying ##f(0)=0## and ##\int_0^1 f'(x)^2 dx\leq 1##, then ##\int_0^1 f(x) dx\leq A.## If possible, find a function that gives equality. (IR)

**8.**(solved by @Antarres ) Let ##n## be a positive integer. Show that ##\binom{2n}{n}## is a multiple of ##4## if and only if ##n## is not a power of ##2##. (IR)

**9.**Show that there are no integer solutions to ##x^3=y^2+21.## (IR)

**10.**Let ##f:S^5\to S^3\times S^2## be a surjective smooth map. Show that ##f## has critical values on every "slice" ##S^3\times\{p\}## for ##p\in S^2.## (IR)

**High Schoolers only**

11.(solved by @Adesh ) Prove that if for ##x\in \mathbb{R}-\{0\}## the number ##x+\dfrac{1}{x}## is an integer, then ##x^n+\dfrac{1}{x^n}## with ##n\in \mathbb{N}## are integers, too.

11.

**12.**(solved by @Not anonymous ) We define for positive integers ##a,b## the following sequence

$$

x_n :=\begin{cases}1 & \text{ if }n=1 \\ ax_{n-1}+b& \text{ if }n>1\end{cases}

$$

Show that the sequence contains infinitely many numbers. which are not prime, for any choice of ##a,b\,.##

**13.**(solved by @Not anonymous ) Name a convergent series ##\sum_{k=1}^\infty a_k## with positive ##a_k##, where ##a_{k+1}/a_k \geq 2## holds infinitely often.

**14.**(solved by @Not anonymous ) For natural numbers ##1\leq k\leq 2n## show that

$$

\binom{2n+1}{k-1}+\binom{2n+1}{k+1} \geq 2\cdot \dfrac{n+1}{n+2}\cdot \binom{2n+1}{k}

$$

**15.**The year on the Earth-like planet Trappist-1e has ##365## days divided into months of ##28,30,31## days. How many months does its year have and how many months with (i) ##28##, (ii) ##30##, (iii) ##31## days?

**16.**(solved by @Not anonymous ) Extra Puzzle. Decrypt the affine encrypted "Ara gtynd hdm hvcrsnd mthvtjph!"

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