# Thread: Calculus & Analysis Marathon & Questions

1. ## Re: First Year Uni Calculus Marathon

Here is one instructive way to do it (probably not the fastest way):

First choose more natural coordinates, u_n=x_n-1, v_n=y_n-2 centred at the fixed point.

Our recurrence then transforms to:

$\begin{pmatrix}u_{n+1}\\ v_{n+1}\end{pmatrix}=\begin{pmatrix}1-2/n & -1/n\\ -1/n & 1-1/n\end{pmatrix}\begin{pmatrix}u_n\\ v_{n}\end{pmatrix}.$

The matrix in question is symmetric, which implies that the eigenvalues are real and the eigenspaces are orthogonal. (An easy computation shows that there are indeed two distinct eigenvalues.)

A consequence of this fact is that these matrix satisfies

$\|Az\|\leq|\lambda_{max}|\|z\|.$
(Exercise).

The maximal (maximal means maximal in absolute value btw) eigenvalue of the n-th matrix is

$\left(1-\frac{3-\sqrt{5}}{n}\right)$

and so convergence of (u_n,v_n) to 0 is proven by showing that the product of these maximal eigenvalues tends to zero, which I am sure you are capable of.

(*) Note that this method establishes convergence for ANY choice of initial point (x_0,y_0). Note also that this is quite related to my most recently posted problem in the linear algebra marathon, which addresses sequences like this but for a fixed multiplication matrix (which is not specified and need not be symmetric).

2. ## Re: First Year Uni Calculus Marathon

$\noindent Prove, for a function that is Riemann Integrable on $0,1$ , \\\\ \lim_{n \to \infty} \sum_{k=1}^n f\left(\frac{k}{n}\right) \varphi(k) = \frac{6}{\pi^2} \int_0^1 x f(x) \text{d}x$

$\noindent Where \varphi is the Euler Totient Function.$

Bonus: Come up with a general statement that extends to any arithmetical function that has a sufficiently "nice" asymptotic approximation.

3. ## Re: First Year Uni Calculus Marathon

$\noindent Prove, for a function that is Riemann Integrable on $0,1$ , \\\\ \lim_{n \to \infty} \sum_{k=1}^n f\left(\frac{k}{n}\right) \varphi(k) = \frac{6}{\pi^2} \int_0^1 x f(x) \text{d}x$

$\noindent Where \varphi is the Euler Totient Function.$

Bonus: Come up with a general statement that extends to any arithmetical function that has a sufficiently "nice" asymptotic approximation.
$Let\quad \beta \quad be\quad a\quad real\quad positive\quad real\quad number$$Let\quad { a }_{ n }\quad be\quad a\quad sequence\quad of\quad positive\quad real\quad numbers\quad such\quad that:\\ \lim _{ n\xrightarrow { } \infty }{ \frac { 1 }{ { n }^{ \beta } } } \sum _{ k=1 }^{ n }{ { a }_{ k } } =L\quad for\quad some\quad L\\ For\quad every\quad continuous\quad function\quad f\quad on\quad \left[ 0,1 \right] ,\\ \lim _{ n\xrightarrow { } \infty }{ \frac { 1 }{ { n }^{ \beta } } } \sum _{ k=1 }^{ n }{ f(\frac { k }{ n } ){ a }_{ k } } =L\int _{ 0 }^{ 1 }{ \beta { x }^{ \beta -1 }f(x)\quad dx } \\ Now\quad we\quad know\quad \\ \lim _{ n\xrightarrow { } \infty }{ \frac { 1 }{ { n }^{ 2 } } } \sum _{ k=1 }^{ n }{ \varphi (k) } =\frac { 3 }{ { \pi }^{ 2 } } \\ \therefore \lim _{ n\xrightarrow { } \infty }{ \frac { 1 }{ { n }^{ 2 } } } \sum _{ k=1 }^{ n }{ f(\frac { k }{ n } )\varphi (k) } =\frac { 6 }{ { \pi }^{ 2 } } \int _{ 0 }^{ 1 }{ { x }f(x)\quad dx } \quad as\quad L=\frac { 3 }{ { \pi }^{ 2 } } and\quad \beta =2$

4. ## Re: Calculus & Analysis Marathon & Questions

$Prove that there exists a unique \emph{non-negative} sequence (x_n) with x_0=0 and n=x_n(x_{n-1}+x_n+x_{n+1}) for n>0.$

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