if-pi-is-rational-then-there-exists-a-I-n-v-2n-n-0-pi-x-n-x-pi-n-sin-x-dx-can-someone-give-a-easier-way-to-expaned-this-

Question Number 60621 by fjdjdcjv last updated on 22/May/19

i f π i s r a t i o n a l t h e n t h e r e

e x i s t s a I_{n} = \frac{v^{2 n}}{n!} \underset{0}{\int^{π}} x^{n} {(x - π)}^{n} s i n (x) d x

c a n s o m e o n e g i v e a e a s i e r w a y t o e x p a n e d t h i s

Commented by MJS last updated on 23/May/19

I {don}^{'} t understand . we can solve the integral

for any given n \in N even if π \notin Q . and what is v ?

Commented by fjdjdcjv last updated on 23/May/19

π = \frac{u}{v}

Commented by MJS last updated on 23/May/19

ok . still the integral exists, quite apart from

the nature of π

another question : \sin x with which period,

2 π or 2 \frac{u}{v} ?

is it \frac{v^{2 n}}{n!} \underset{0}{\int^{\frac{u}{v}}} x^{n} (x - \frac{u}{v}) \sin \frac{v π x}{u} d x ?

or shall we calculate I_{n} using the usual π

and then show that π \notin Q ?

Commented by MJS last updated on 23/May/19

sorry this is quite confusing me

Commented by MJS last updated on 23/May/19

I_{1} = - 4 v^{2}

I_{2} = 2 v^{4} (- π^{2} + 12)

I_{3} = 24 v^{6} (π^{2} - 10)

I_{4} = 2 v^{8} (π^{4} - 180 π^{2} + 1680)

I_{5} = 60 v^{10} (- π^{4} + 112 π^{2} - 1008)

\dots

all calculated with the usual π

if π = \frac{u}{v}

I_{1} = - 4 v^{2}

I_{2} = - 2 (u^{2} - 12 v^{2}) v^{2}

I_{3} = 24 (u^{2} - 10 v^{2}) v^{4}

I_{4} = 2 (u^{4} - 180 u^{2} v^{2} + 1680 v^{4}) v^{4}

I_{5} = - 60 (u^{4} - 112 u^{2} v^{2} + 1008 v^{4}) v^{6}

\dots

but what do we know now ?

Facebook Tweet Pin