for integers,
= 
= 
, as 
the 
which gets us the 
form therefore L'hopital came to my mind , which might be a completely wrong intuition, anyway what should i do to find that limit
be defined by 
(a) Show that 
has exactly one point of discontinuity.
![$f \in C_{\text{per}}([-\pi,\pi]; \mathbb{C})$](http://latex.artofproblemsolving.com/7/a/a/7aa8f551ba6422af33ab978aaf842329b6c2aced.png)
such that its Fourier series diverges at 
.![$t \in [-\pi,\pi]$](http://latex.artofproblemsolving.com/d/1/3/d1331fab5626e64ec75c6c40edc6379bc591c803.png)
,![$f_t \in C_{\text{per}}([-\pi,\pi]; \mathbb{C})$](http://latex.artofproblemsolving.com/e/1/e/e1ef8ac797a9743ebd9e71bf1682b0246b221086.png)
such that the Fourier series of 
diverges at 
,
be a Banach space and for every 
we have a normed space 
.
and 
such that![\[
\sup_{k \geq 1} \| T_{n,k}x_n \|_{Y_n} = \infty.
\]](http://latex.artofproblemsolving.com/a/1/b/a1b7a2453200f95f61c2451fc91a8fc3a81fd8fd.png)
Show that![\[
B = \left\{ x \in X : \sup_{k \geq 1} \| T_{n,k}x \|_{Y_n} = \infty \ \forall n \geq 1 \right\}
\]](http://latex.artofproblemsolving.com/4/1/a/41a671b6c40492f454bf79d1a3205cd5592a94e6.png)
is of second category. (I am given the hint to write 
as![\[
A = \bigcup_{n \geq 1} A_n = \bigcup_{n \geq 1} \left\{ x \in X : \sup_{k \geq 1} \| T_{n,k}x \|_{Y_n} < \infty \right\}
\]](http://latex.artofproblemsolving.com/b/9/1/b9174ec68dd2a503c6d68deaf281c5e110637cbd.png)
and show that 
is of first category.)
.![$f_D \in C_{\text{per}}([-\pi,\pi]; \mathbb{C})$](http://latex.artofproblemsolving.com/8/a/b/8ab83356343e4c5e30778770377027d0b2d57735.png)
such that the Fourier series of 
diverges at 
for all ![\[
T_{n,k} : (C_{\text{per}}([-\pi,\pi]; \mathbb{C}), \| \cdot \|_\infty) \to \mathbb{C}, \quad f \mapsto S_k(f)(t_n),
\]](http://latex.artofproblemsolving.com/2/2/5/225a5eec7a9582981841918fc28de96379b31404.png)
where![\[
S_k(f)(x) = \sum_{|j| \leq k} c_j(f) e^{ijx}.
\]](http://latex.artofproblemsolving.com/a/c/3/ac37ad818424867f4504e1472b2abe0883b6284b.png)
and 
then is this uniformly convergence on 
comment on uniformly convergence on ![$[0,1]$](http://latex.artofproblemsolving.com/e/8/6/e861e10e1c19918756b9c8b7717684593c63aeb8.png)
where in general it is should be uniformly convergence.
and trying to solv
be a 
-
and 
,
and 
is defined as ![\[\langle v, w \rangle = \sum_{i=1}^{10} v_i w_i.\]](http://latex.artofproblemsolving.com/9/f/9/9f9b7ff2fb19fee7e935c1b18dea3ec80547b9f9.png)
For 
,
on ![\[ P_u : V \to V, \quad x \mapsto x - \frac{2\langle x, u \rangle u}{\langle u, u \rangle} \]](http://latex.artofproblemsolving.com/9/6/4/9646290e2b5e540b7f85d0d71e4d8992ccd8f542.png)
Given 
satisfying![\[ 0 < \langle v, w \rangle < \sqrt{\langle v, v \rangle \langle w, w \rangle} \]](http://latex.artofproblemsolving.com/d/8/0/d800fe48958cfee9ac01aa8bf11fe431ef45f8ef.png)
let 
.
satisfying 
is 
into a 3-year certificate of deposit that earned 10% annual interest, compounded annually. Audrey made no additional deposits to or withdrawals from the certificate of deposit. What was the value of the certificate of deposit at the end of the 3-year period?
be a circle 
with counterclockwise orientation and 
be a circle 
with clockwise orientation.
,
when t tend 
:
which pass through the point 
Y by teomihai
Let 
a real number. Given a differentiable function 
, let 
the curve 
on the XY-plane. For all point 
on , the tangent to on intersect the x-axis and the y-axis on 
and 
, respectively, such 
and 
.
If
belongs to , find all possible functions 
.
a real number. Given a differentiable function 
, let 
the curve 
on the XY-plane. For all point 
on , the tangent to on intersect the x-axis and the y-axis on 
and 
, respectively, such 
and 
.If
belongs to , find all possible functions 
.This post has been edited 1 time. Last edited by KAME06, Apr 8, 2025, 5:07 AM
be a point on 
such that the tangent line at ![$y-f(t)=f'(t)(x-t) \Rightarrow y =f'(t) \cdot x + [f(t)-tf'(t)]$](http://latex.artofproblemsolving.com/7/a/9/7a969f5d2efafdc65e3c8ac920d3fcaa8abfaa63.png)
(i);
and 
i
(ii). If 
for 
,
;![$t^2f'(t)^{2} + t^2f'(t)^{4} = \alpha[f(t)^{2}f'(t)^{2} + f(t)^{2}]$](http://latex.artofproblemsolving.com/2/6/f/26fc21fa2cbfad9fe9f49117ae52551dcf1e1d81.png)
;![$f'(t)^{2} = \frac{[\alpha^{2}f(t)^{2}-t^2] \pm \sqrt{[\alpha^{2}f(t)^{2}-t^2]^2 + 4[\alpha t f(t)]^2}}{2t^2}$](http://latex.artofproblemsolving.com/6/d/7/6d7c8714b01875d010c8b74e70bec0f4b725c717.png)
;![$f'(t)^{2} = \frac{[\alpha^{2}f(t)^{2}-t^2] \pm \sqrt{[t^2+\alpha^{2}f(t)^{2}]^2}}{2t^2} = \frac{[\alpha^{2}f(t)^{2}-t^2] \pm [t^2+\alpha^{2}f(t)^{2}]}{2t^2}$](http://latex.artofproblemsolving.com/f/c/c/fcce354600e35d2baa78ac0acf793a9eeffe95f1.png)
;
(iii).
is our initial condition, then we obtain:
(iv);
and 
.
