- #1
tripsynth
- 3
- 0
Hi, I'm curious in learning calculus again but through a more rigorous fashion than plug and chug that I did last year.
I'm trying to go slowly through Apostol's calculus but the way he's deriving the integral for a parabolic function x^2 is kinda troubling me.
I'm fine until he reaches a point where he wants to prove that
1^2 + 2^2 + ... + n^2 = n^3/3 +n^2/2 + n/6
he ends up proving it by examining
(k+1)^3 = k^3 + 3k^2 + 3k + 1
my problem is how did he decide to use (k+1)^3?
one more issue is he later goes on to state
1^2 + 2^2 + ... + (n-1)^2 < n^3/3 < 1^2 + 2^2 + ... + n^2
and has a proof in the later sections for it.
where did the n^3/3 come from ?? why is he using it?
I feel like I'm missing something in my math education. Should I go through a number theory book before continuing ?
I'm trying to go slowly through Apostol's calculus but the way he's deriving the integral for a parabolic function x^2 is kinda troubling me.
I'm fine until he reaches a point where he wants to prove that
1^2 + 2^2 + ... + n^2 = n^3/3 +n^2/2 + n/6
he ends up proving it by examining
(k+1)^3 = k^3 + 3k^2 + 3k + 1
my problem is how did he decide to use (k+1)^3?
one more issue is he later goes on to state
1^2 + 2^2 + ... + (n-1)^2 < n^3/3 < 1^2 + 2^2 + ... + n^2
and has a proof in the later sections for it.
where did the n^3/3 come from ?? why is he using it?
I feel like I'm missing something in my math education. Should I go through a number theory book before continuing ?