Question

Orthogonality of two vectors of infinite dimension.

Answer 1

How would you show that a vector containing successive alternating even powers of (pi/2) is orthogonal to a vector containing only the inverse of even factorials?

Answer 2

\[\Large <(\frac{\pi}{2})^0, -(\frac{\pi}{2})^2,(\frac{\pi}{2})^4,...>\] and \[\Large <\frac{1}{0!}, \frac{1}{2!},\frac{1}{4!},...>\]

show that these are orthogonal

Answer 3

mmm any trick with dot product ?
and we might have some series ?

Answer 4

You are on the right path! I just came up with this for fun, and thought it was an interesting thing to consider and wanted to see how people would approach this problem haha. =D

Answer 5

lol ok ill try
but u shouldnt give me medal untill i get it :P

Answer 6

Ok good luck. I'll give a hint, if I replaced pi/2 with n, there are infinitely many values you can pick for which the two vectors are orthogonal, and there is a formula for all possible n's that make them orthogonal.

That might either make it more confusing or help, I don't know haha.

Answer 7

so we wanna prove hehe xD

\(\Huge \sum_{\large n=1}^{ \large \infty} \left ( (-1)^n (\frac{ \pi }{2})^{\large 2n}.\frac{1}{(2n)!} \right )=0\)

Answer 8

Haha pretty much. Except the index starts at n=0 heh. =P

Answer 9

hehe its ok lol small typo
\(\Huge \sum_{\large n=0}^{ \large \infty} \left ( (-1)^n (\frac{ \pi }{2})^{\large 2n}.\frac{1}{(2n)!} \right )=0 \)

Answer 10

idk why i wanna use some fourier for some function :o

Answer 11

You're on the right path kind of, but you might be overcomplicating it a little hehe.

Answer 12

haha well , its couse i cant think in other place other than my home xD

Answer 13

I know what you mean! I guess you'll just have to have this bothering you the rest of the day until you figure it out! =P

Answer 14

Would it suffice to show that the partial sums approach 0? You essentially are proving a limit, so you could try satisfying Cauchy conditions for convergence.

Answer 15

hint: Taylor Series.

Answer 16

hehe finally lol
cos pi/2 :P

Answer 17

wew !

Answer 18

XD Nice!

Answer 19

:P
hehe

Answer 20

Fun, I wonder what interesting or useful things you can get out of considering other Taylor series as infinite dot products I wonder? =P

Answer 21

wonder as you want :P