Find the sum of the series:
I do the sum and each time I get a different answers, none of them matching with the answer in my book.
Book's answer:-n(2n+1)
My latest answer: -3n(2n-1)
My latest working:
(a)
starting from r=1, there is (n-1) terms and 1^2 have to be added separately.
(b)
starting from r=1, there is n terms.
----
(a)
(b)
Summation=(a)-(b)=-6n^2+3n
=-3n(2n-1)
Can you please narrow down the field where I should search the error?
I am exasperated looking through my rough handwriting.
Okay, I'll try to explain !
You first wrote
But this equals :
This is supposed to equal
So the red part is repeating !
Moreover, the last term isn't equal to 2n-1 but to 2n-3. So in fact, the correct formula is :
Similarly,
____________________________________
The sum S you're looking for is :
The blue stuff simplifies to 0.
So
Is it clearer ?
I wrote
" alt="1+\sum_{r=1}^{n-1} (2r-1)^2
" /> here but did the sum first time with this
" alt="1+\sum_{r=1}^{n-1} (2r+1)^2
" />
So I retained r=1 in 2r+1=3 and the series goes when I switched to (2r-1)^2 second time when I did the sum.
My head is feeling light, I wonder why