Thread: proving the summation of a finite series .

1. proving the summation of a finite series .

FInd $\sum^{n}_{r=1}r^3$

My attempt :

Using this identity : $r^4-(r-1)^4=4r^3-6r^2+4r-1$

$\sum^{n}_{r=1}r^4-(r-1)^4=\sum^{n}_{r=1}4r^3-6r^2+4r-1$

The LHS can be written as $n^4$ .

$4\sum^{n}_{r=1}r^3=n^4+n(n+1)(2n+1)+2n(n+1)-n$

but it is not anywhere nearer to $\frac{1}{4}n^2(n+1)^2$ ,there must be some problem with my proving .

FInd $\sum^{n}_{r=1}r^3$

My attempt :

Using this identity : $r^4-(r-1)^4=4r^3-6r^2+4r-1$

$\sum^{n}_{r=1}r^4-(r-1)^4=\sum^{n}_{r=1}4r^3-6r^2+4r-1$

The LHS can be written as $n^4$ .

$4\sum^{n}_{r=1}r^3=n^4+n(n+1)(2n+1)+2n(n+1)-n$

but it is not anywhere nearer to $\frac{1}{4}n^2(n+1)^2$ ,there must be some problem with my proving .
You have made a couple of sign errors. The last line should be:

$4\sum^{n}_{r=1}r^3 = n^4+n(n+1)(2n+1) {\color{red} - } 2n(n+1) {\color{red} + } n$

and the right hand side obligingly factorises into $n^2 (n+1)^2$.

3. Thanks Mr F , i try to factorise and i still can't get the proof right .

$
4\sum^{n}_{r=1}r^3 = n^4+n(n+1)(2n+1)-2n(n+1)+n
$

I'll modify your work ... and include all the steps.

FInd: . $\sum^{n}_{r=1}r^3$

Using this identity: . $r^4-(r-1)^4\:=\:4r^3-6r^2+4r-1$

. . we have: . $\sum^{n}_{r=1}\bigg[r^4-(r-1)^4\bigg]\;=\;\sum^{n}_{r=1}\bigg[4r^3-6r^2+4r-1\bigg]$

The LHS can be written as $n^4$ .

We have: . $n^4 \;=\;4\sum r^3 - 6\sum r^2 + 4\sum r - \sum 1$

. . . . . . . $n^4 \;=\;4\sum r^3 - 6\!\cdot\!\tfrac{n(n+1)(2n+1)}{6} + 4\!\cdot\!\tfrac{n(n+1)}{2} - n$

. . . . . . . $n^4 \;=\;4\sum r^3 - n(n+1)(2n+1) + 2n(n+1) - n$

$\text{We have: }\;4\sum r^3 \;=\;\underbrace{n^4 + n}_{\quad\searrow} + n(n+1)(2n+1) - 2n(n+1)$

. . . . . . $4\sum r^3 \;=\;\overbrace{n(n+1)(n^2-n+1)} + n(n+1)(2n+1) - 2n(n+1)$

Factor: . . $4\sum r^3 \;=\;n(n+1)\bigg[(n^2-n+1) + (2n+1) - 2\bigg]$

. . . . . . . $4\sum r^3 \;=\;n(n+1)\bigg[n^2+n\bigg] \;=\;n(n+1)\cdot n(n+1) \;=\;n^2(n+1)^2$

Therefore: . $\sum^n_{r=1}r^3 \;=\;\frac{1}{4}n^2(n+1)^2$

5. Thanks a lot Soroban , btw

$n^4+n=n(n+1)(n^2-n+1)$ , is it an algebraic identity ?

6. Yes. Use the fact that: ${\color{red}a^3 + b^3} = {\color{blue}(a+b)(a^2 - ab + b^2)}$

So: $n^4 + n \ = \ n({\color{red}n^3 + 1}) \ = \ n {\color{blue}(n+1)(n^2 - n + 1)}$