i made a typing mistake
there is no absolute value in the expression
and i still cant see the logic in this separation
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
or we could have (if you want the absolute value in the exponent of on the left):
Now if you could clarify what the sum on the left is supposed to be and what you are trying to do, that would help. At the moment I feel I am wasting my time as you have not ma
de it clear what you are trying to do or why.
Yes, it is- but notice that Captain Black has switched indices also.
The sum in question is . Since n is negative, that is the same as . Now let k= -n so that n= -k, when n= -1, k= 1 and when , . In terms of index k the sum is .
Finally, since this is a "dummy index" anyway, it doesn't matter whether you call it "k" or "n". That is exactly the same as
If n+ 3= 6, I can replace n by "-k" and get -k+ 3= 6.
I could then replace k by n and get -n+ 3= 6. The two equations "n+3= 6" and "-n+ 3= 6" [b]would[b] contradict each other if they were the same n.
But in sums we have, as I said "dummy" indices, if , I can replace n by -k and have and then replace k by n to get .
Those all say the same thing: .