i want to show the sum of odd is zero

[logistic_guy]

here is the question

Show that if $\displaystyle x[n]$ is an odd signal, then $\displaystyle \sum_{n=-\infty}^{\infty} x[n] = 0$.

is it correct to say if $\displaystyle x[n] = n^3$ then the partial sum is $\displaystyle \sum_{n=-1}^{1} x[n] = \sum_{n=-1}^{1} n^3 = -1 + 0 + 1 = 0$ then $\displaystyle \sum_{n=-\infty}^{\infty} n^3 = 0$

 

[lev888]

here is the question

Show that if $\displaystyle x[n]$ is an odd signal, then $\displaystyle \sum_{n=-\infty}^{\infty} x[n] = 0$.

is it correct to say if $\displaystyle x[n] = n^3$ then the partial sum is $\displaystyle \sum_{n=-1}^{1} x[n] = \sum_{n=-1}^{1} n^3 = -1 + 0 + 1 = 0$ then $\displaystyle \sum_{n=-\infty}^{\infty} n^3 = 0$

I would rearrange the sum into a sum of pairs, each of which is 0 by definition.

 

[Dr.Peterson]

here is the question

Show that if $\displaystyle x[n]$ is an odd signal, then $\displaystyle \sum_{n=-\infty}^{\infty} x[n] = 0$.

is it correct to say if $\displaystyle x[n] = n^3$ then the partial sum is $\displaystyle \sum_{n=-1}^{1} x[n] = \sum_{n=-1}^{1} n^3 = -1 + 0 + 1 = 0$ then $\displaystyle \sum_{n=-\infty}^{\infty} n^3 = 0$

One example is not a proof; but with some extra words, this example could be used to "show" (informally) why it generalizes to any odd function and infinitely many terms (assuming convergence).

In other words, what @lev888 said!

 

[blamocur]

assuming convergence

Depends on how we define $\sum_{-\infty}^\infty$, doesn't it? I.e., if $\sum_{-\infty}^\infty = \lim_{k\rightarrow\infty} \sum_{-k}^k$ then we don't depend on convergence of odd signals/sequences.

 

[bamboum]

One can use generalized function of Scharwz theory. Int of comb (x)*x will be the integation of odd function on symmetric interval. That is 0.
Comb is or cha(x) is a sum of Dirac's distributions

 

[logistic_guy]

thank you guys

is it correct to write this notation?

$\displaystyle \sum_{n=-\infty}^{\infty} x[n] = \sum_{n=-\infty}^{\infty} n^3 = .... + (-n)^3 + .... + (-5)^3 + .... + 0^3 + .... + 5^3 + .... + n^3 + .... = .... + (5^3 - 5^3) + .... + (n^3 - n^3) + .... = 0$

maybe i can do this

$\displaystyle \sum_{n=-\infty}^{\infty} x[n] = ...... + x[-n] + ....... + x[0] + ........ + x[n] + ........ = ..... + (x[-n] + x[n]) + ....... = 0$

One can use generalized function of Scharwz theory. Int of comb (x)*x will be the integation of odd function on symmetric interval. That is 0.
Comb is or cha(x) is a sum of Dirac's distributions

when i answer this question correctly, write your full solution. you can now write some of it with mathematics notation

 

[blamocur]

is it correct to write this notation?

I don't see any problem with it, but notation with ellipsis is usually somewhat ambiguous.

 

[logistic_guy]

I don't see any problem with it, but notation with ellipsis is usually somewhat ambiguous.

thank blamocur