Apostol volume 1 calc? what do I need

In summary, Apostol is trying to derive the integral for a parabolic function, but is having trouble with a particular step. He uses the n^3/3 to help prove the upper and lower sums converge.
  • #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 ?
 
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  • #2
well that's something practice would teach u

i suggest u better go through some elementary induction problems , that should do
 
  • #3
tripsynth said:
(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?

The reason he uses (k+1) ^3 is to set up a telescoping series:(k+1)^3 - k^3= 3k^2 + 3k + 1. the idea is to get an expression for the sum of squares.

(2)^3 - (1)^3 + (3)^3 - (2)^3 + ... (k+1)^3 - (k)^3 = (k+1)^3 -1 (as all the other terms cancel). Do you see how this gives (K+ 1) ^3 - 1 = 3(1^2 + 2^2 + ... + k^2) + 3(1 + ... + k) + k? Since we know what 3(1 + ... +k) is we can now solve for (1^2 + ... + k^2).

You've taken calculus so you know the integral of x^2 is (x^3)/3. Apostal is trying to derive this formula with the fundamental theorem of calculus. That's where the n^3/3 comes from. The point of the inequality is to use the squeeze theorem to prove the upper and lower sum (Riemann sums or inf/sup definition I don't know what Apostal uses) converge to (x^3)/3 as n-> infinity. Do you understand the later parts of his derivation?
 
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  • #4
deluks917 said:
The reason he uses (k+1) ^3 is to set up a telescoping series:(k+1)^3 - k^3= 3k^2 + 3k + 1. the idea is to get an expression for the sum of squares.

(2)^3 - (1)^3 + (3)^3 - (2)^3 + ... (k+1)^3 - (k)^3 = (k+1)^3 -1 (as all the other terms cancel). Do you see how this gives (K+ 1) ^3 - 1 = 3(1^2 + 2^2 + ... + k^2) + 3(1 + ... + k) + k? Since we know what 3(1 + ... +k) is we can now solve for (1^2 + ... + k^2).

You've taken calculus so you know the integral of x^2 is (x^3)/3. Apostal is trying to derive this formula with the fundamental theorem of calculus. That's where the n^3/3 comes from. The point of the inequality is to use the squeeze theorem to prove the upper and lower sum (Riemann sums or inf/sup definition I don't know what Apostal uses) converge to (x^3)/3 as n-> infinity. Do you understand the later parts of his derivation?

I'll look over the telescoping series and read over that section again later.

yeah I knew that integral but on that page it just appears with that inequality. he doesn't develop n^3/3 it just appears at that moment. I think I understand the rest of the proof though. I just was confused about where n^3/3 came from.

Thank you for your help.
 

1. What is "Apostol volume 1 calc"?

"Apostol volume 1 calc" refers to the first volume of the textbook "Calculus" written by mathematician and educator Tom M. Apostol. It covers topics such as limits, derivatives, and integrals.

2. Is "Apostol volume 1 calc" suitable for beginners in calculus?

It is recommended for students who have a strong foundation in algebra and trigonometry. It may be challenging for beginners, but it provides a thorough and rigorous approach to calculus.

3. What topics are covered in "Apostol volume 1 calc"?

The book covers topics such as limits, continuity, derivatives, applications of derivatives, and integrals. It also includes exercises and examples to help readers understand the concepts.

4. Do I need any prior knowledge before using "Apostol volume 1 calc"?

As mentioned earlier, it is recommended to have a strong understanding of algebra and trigonometry before using this textbook. It may also be helpful to have some knowledge of basic calculus concepts.

5. Are there any supplemental materials needed for "Apostol volume 1 calc"?

The textbook itself is comprehensive and does not require any additional materials. However, some students may find it helpful to use online resources or a calculator while working through the exercises.

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