# Thread: Series Concept

1. ## Series Concept

Hey guys

I'm trying to teach myself about the topic series but can't seem to understand it...

Esp. these questions:

1. (x+1)^3 – x^3 = 3x^2 + 3x^ + 1
∑n^2 =

2. (x+1)^4 – x^4 = 4x^3 + 6x^2 + 4x + 1
∑n^3=

Please, culd u please explain how to do these q.s to me??

Thanx a lot!

2. ummm

For question 1, some of my working out is:

(x+1)^3 - x^3 = 3x^2 + 3x + 1

x = 1 2^3 - 1^3 = 3 * 1^2 + 3 * 1 + 1
x = 2 3^3 - 2^3 = 3* 2^2 + 3 * 2 + 1

x = n (n+1)^3 - n^3 = 3 * n^3 - n^3 = 3 * n^2 + 3 * n + 1

(n + 1 - 1)((n+1)^2 + (n+1) + 1^2) = 3En^3 + (3*n(n+1))/2 + n

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Idno ...somehow we're meant to get En^2 = n/6(n+1)(2n+1) but I don't really noe the rite method to get to it

3. Hi

$\displaystyle (k+1)^3 - k^3 = 3 k^2 + 3 k + 1$

Sum for k from 0 to n
$\displaystyle (n+1)^3 - 0^3 = 3 \sum_{k=0}^n k^2 + 3 \sum_{k=0}^n k + n+1$

You know that $\displaystyle \sum_{k=0}^n k = \frac{n(n+1)}{2}$

Therefore $\displaystyle 3 \sum_{k=0}^n k^2 = (n+1)^3 - 3 \frac{n(n+1)}{2} - (n+1)$

Then $\displaystyle \sum_{k=0}^n k^2 = \frac{n(n+1)(2n+1)}{6}$

4. Originally Posted by xwrathbringerx
ummm

For question 1, some of my working out is:

(x+1)^3 - x^3 = 3x^2 + 3x + 1

x = 1 2^3 - 1^3 = 3 * 1^2 + 3 * 1 + 1
x = 2 3^3 - 2^3 = 3* 2^2 + 3 * 2 + 1

x = n (n+1)^3 - n^3 = 3 * n^3 - n^3 = 3 * n^2 + 3 * n + 1

(n + 1 - 1)((n+1)^2 + (n+1) + 1^2) = 3En^3 + (3*n(n+1))/2 + n

-------------------------

Idno ...somehow we're meant to get En^2 = n/6(n+1)(2n+1) but I don't really noe the rite method to get to it
You have

$\displaystyle {\left( {n + 1} \right)^3} = {n^3} + 3{n^2} + 3n + 1 \Leftrightarrow 3{n^2} = {\left( {n + 1} \right)^3} - {n^3} - 3n - 1$

Now consider the value $\displaystyle n = 1,{\text{ }}2,{\text{ }}3,{\text{ }}4,{\text{ }} \ldots ,{\text{ }}k$, i.e.

$\displaystyle 3 \cdot {1^2} = {\left( {1 + 1} \right)^3} - {1^3} - 3 \cdot 1 - 1$

$\displaystyle 3 \cdot {2^2} = {\left( {2 + 1} \right)^3} - {2^3} - 3 \cdot 2 - 1$

$\displaystyle 3 \cdot {3^2} = {\left( {3 + 1} \right)^3} - {3^3} - 3 \cdot 3 - 1$

$\displaystyle \begin{gathered}3 \cdot {4^2} = {\left( {4 + 1} \right)^3} - {4^3} - 3 \cdot 4 - 1 \hfill \\\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \hfill \\3 \cdot {k^2} = {\left( {k + 1}\right)^3} - {k^3} - 3 \cdot k - 1 \hfill \\ \end{gathered}$

Now summarizes these values and you will get it

$\displaystyle 3 \cdot \sum\limits_{n = 1}^k {{n^2}} = {\left( {k + 1} \right)^3} - {1^3} - 3 \cdot \sum\limits_{n = 1}^k n - k = {\left( {k + 1} \right)^3} - \frac{3}{2}k\left( {k + 1} \right) - \left( {k + 1} \right) =$

$\displaystyle = \left( {k + 1} \right)\left( {{{\left( {k + 1} \right)}^2} - \frac{3}{2}k - 1} \right) = \left( {k + 1} \right)\left( {{k^2} + \frac{k}{2}} \right) = \frac{{k\left( {k + 1} \right)\left( {2k + 1} \right)}}{2} \Leftrightarrow$

$\displaystyle \Leftrightarrow \sum\limits_{n = 1}^k {{n^2}} = \frac{{k\left( {k + 1} \right)\left( {2k + 1} \right)}}{6}.$