Breaking this expressions into areas.

[nhrock3]

[tex]\sum_{n=-\infty}^{\infty}r^{|n|}e^{inx}=[/tex][tex]\sum_{n=1}^{\infty}e^{inx}[/tex]+ [tex]r^{0}e^{0}[/tex] +[tex]\sum_{n=-\infty}^{\infty}r^{n}e^{-inx}[/tex]

if we look at the left side
i have been told that it was broken by intervals
the left most is for n=1.. infinity the central is for n=0
the right most for n=-1 ..-infinity

i know the definition of the |x| function
for some values its x for other its -x

but here the power of e changes too
why?

 

[tiny-tim]

[tex]\sum_{n=-\infty}^{\infty}r^{|n|}e^{inx}=[/tex][tex]\sum_{n=1}^{\infty}e^{inx}[/tex]+ [tex]r^{0}e^{0}[/tex] +[tex]\sum_{n=-\infty}^{\infty}r^{n}e^{-inx}[/tex]

if we look at the left side
i have been told that it was broken by intervals
the left most is for n=1.. infinity the central is for n=0
the right most for n=-1 ..-infinity

i know the definition of the |x| function
for some values its x for other its -x

but here the power of e changes too
why?

Hi nhrock3!

I suspect you've copied it wrong, is it …

[tex]\sum_{n=-\infty}^{\infty}r^{|n|}e^{inx}\ =\ [/tex] [tex]\sum_{n=1}^{\infty}r^{n}e^{inx}[/tex]+ [tex]r^{0}e^{0}[/tex] +[tex]\sum_{n=1}^{\infty}}r^{n}e^{-inx}[/tex]

?​

 

[nhrock3]

why we always have [tex]r^n[/tex] why in the minus case we don't have [tex]r^{-n}[/tex]

and why the power of the exponent changes to minus in the negative case
?

 

[tiny-tim]

Because it depends what the limits are.

Perhaps it's clearer if I write it …

[tex]\sum_{n=-\infty}^{\infty}r^{|n|}e^{inx}\ =\ [/tex] [tex]\sum_{n=1}^{\infty}r^{n}e^{inx}[/tex]+ [tex]r^{0}e^{0}[/tex] +[tex]\sum_{m=1}^{\infty}}r^{m}e^{-imx}[/tex]

where m = -n, so |n| = m, and inx = -imx.

(and m and n are "dummy" indices, so you can then change m to n )

 

[nhrock3]

i can't see how the minus indexes changes the original expression

?

why the power of the exponent changes to minus in the negative case

they should be negative
and we don't need to change it to minus

 

[HallsofIvy]

Do you see that both sums are for positive n?
Your original sum is
[tex]\sum_{n=-\infty}^\infty r^{|n|}e^{inx}[/tex]

You can break that into three parts, as you say:
[tex]\sum_{n=-\infty}^{-1} r^{|n|}e^{inx}+ r^0e^{i0x}+ \sum_{n=1}^\infty r^{|x|}e^{inx}[/tex]

Now look at each of that first sum separately. For n< 0, |n|= -n so
[tex]\sum_{n=-\infty}^{-1} r^{|n|}e^{inx}= \sum_{n=-\infty}^{-1} r^{-n}e^{inx}[/tex]
Change the index: let k= -n so n= -k:
[tex]= \sum_{-k= -\infty}^{-1} r^{k}e^{i(-k)x}= \sum_{k=1}^\infty r^k e^{-ikx}[/tex]

Since the index here is "dummy" we can use any letter we like, say "n", in place of k:
[tex]= \sum_{n= 1}^\infty r^n e^{-inx}[/tex]

Of course, [itex]r^0e^{i(0)x}= 1[/itex], and, for n> 0, |n|= n, so the entire sum becomes
[itex]\sum_{n= 1}^\infty r^n e^{-inx}+ 1+ \sum_{n=1}^\infty r^n e^{inx}[/tex].

 

[tiny-tim]

i can't see how the minus indexes changes the original expression

Try changing m to -n in the right-hand term in my equation …

what do you get? ​

 

[nhrock3]

no you can't use is as dummy
if n=-k
and k=n
then n=-n

which works only for n=0

 

[nhrock3]

Do you see that both sums are for positive n?
Your original sum is
[tex]\sum_{n=-\infty}^\infty r^{|n|}e^{inx}[/tex]

You can break that into three parts, as you say:
[tex]\sum_{n=-\infty}^{-1} r^{|n|}e^{inx}+ r^0e^{i0x}+ \sum_{n=1}^\infty r^{|x|}e^{inx}[/tex]

Now look at each of that first sum separately. For n< 0, |n|= -n so
[tex]\sum_{n=-\infty}^{-1} r^{|n|}e^{inx}= \sum_{n=-\infty}^{-1} r^{-n}e^{inx}[/tex]
Change the index: let k= -n so n= -k:
[tex]= \sum_{-k= -\infty}^{-1} r^{k}e^{i(-k)x}= \sum_{k=1}^\infty r^k e^{-ikx}[/tex]

Since the index here is "dummy" we can use any letter we like, say "n", in place of k:
[tex]= \sum_{n= 1}^\infty r^n e^{-inx}[/tex]

Of course, [itex]r^0e^{i(0)x}= 1[/itex], and, for n> 0, |n|= n, so the entire sum becomes
[itex]\sum_{n= 1}^\infty r^n e^{-inx}+ 1+ \sum_{n=1}^\infty r^n e^{inx}[/tex].

no you can't use it as dummy
if n=-k
and k=n
then n=-n

which works only for n=0
and for n= -1 to -infinity

 

[HallsofIvy]

no you can't use it as dummy
if n=-k
and k=n
then n=-n

which works only for n=0
and for n= -1 to -infinity

What that tells me is that you don't know what a "dummy" index is. If I had an equation that said k+ n= 3, then I could not replace k by either n or -n- they mean different things. But if my equation is [itex]\sum_{k=1}^5 \frac{1}{k}+ \sum{n=1}^5 \frac{(-1)^n}{n}[/itex] I could replace the "k" in the first sum by "n" because [itex]\sum_{k=1}^5 \frac{1}{k}= 1+ \frac{1}{2}+ \frac{1}{3}+ \frac{1}{4}+ \frac{1}{5}[/itex] and [itex]\sum_{n=1}^5 \frac{1}{n}= 1+ \frac{1}{2}+ \frac{1}{3}+ \frac{1}{4}+ \frac{1}{5}[/itex] are both numbers and do not depend upon either "n" or "k". Having changed from "n" to "k", I could now make the change from k to -n as well and have [itex]\sum_{n=-1}^{-5}\frac{1}{-n}= \frac{1}{-(-1)}+ \frac{1}{-(-2)}+ \frac{1}{-(-3)}+ \frac{1}{-(4)}+ \frac{1}{-(-5)}[/itex] which is exactly the same thing. This "n" has nothing to do with the original n. That does NOT imply "k= n" and "k= -n" for the same n!

 

[nhrock3]

aahhh thanks :)