Sums of Powers of Natural Numbers
>2 We'll use the symbol W55 for the sum of the powers of the first 8 natural numbers. In other words,
55 5 Wœ"#ÞÞÞ8Þ5
Of course, this is a “formula” for W5 , but it doesn't help you compute it doesn't tell you how to find the $$ $ exact value, say, of Wœ"#ÞÞÞ"&$ . We'd like to get what's called a closed formula for W5 , that is, one without the annoying “ ... ” in it.
!! ! For Wœ"#ÞÞÞ8! , this is easy: since there are 8 terms, each equal to " , so we get
W! œ""ÞÞÞ" œ"†8 œ8
For WW" , it's already harder. Here's a slick way of finding a closed formula for " :
Write down W" twice, in two different orders:
W" œ " # $ ÞÞÞ Ð8 "Ñ 8 , and W" œ 8 Ð8 "Ñ Ð8 #Ñ ÞÞÞ # " Then add to get: ______#W" œ Ð8 "Ñ Ð8 "Ñ Ð8 "Ñ ... + Ð8 "Ñ Ð8 "Ñ .
Since there are 8Ð8"Ñß terms on the right, each equal to we get
#W" œ 8Ð8 "Ñ , so
8Ð8"Ñ Wœ" #
"&Ð"'Ñ This is a “usable” closed formula: for example, "#$ÞÞÞ"&œ# œ"#! .
Here's a list of formulas for:
!! ! Wœ"#ÞÞÞ8œ8!
"" "8Ð8"Ñ Wœ"#ÞÞÞ8œ" #
## #8Ð8"ÑÐ#8"Ñ Wœ"#ÞÞÞ8œ# '
8Ð8"Ñ ## Wœ$$" [# ] (Curious observation: WœW [ ] )
Where do these formulas come from? There is a systematic way to get a formula for each W5 once you know the previous formulas for W!" ß W ß ÞÞÞß W 5" À We'll illustrate here with just two examples:
!! ! i) If someone noticed the (easy) fact the W! œ" # ÞÞÞ8 œ""ÞÞÞ"œ8 "" " how could this fact be used to get a formula: W" œ" # ÞÞÞ8 œ "#ÞÞÞ8œ ???
Well, for any positive integer 4 , we know that Ð4"Ñ## 4 œ#4" . So we write this down substituting in each value 4 œ "ß 4 œ #ß ÞÞÞß 4 œ 8
### " œ #Ð"Ñ " $## # œ #Ð#Ñ " %## $ œ #Ð$Ñ " ã Ð8"Ñ## 8 œ#Ð8Ñ" Adding up the columns on both sides (with lots of cancellations on the left-hand side) gives ______Ð8"Ñ# " œ#Ð"#ÞÞÞ8Ñ8 œ# W" 8 so # 8 #8""œ#W" 8 so # 88œ#W" so 88# 8Ð8"Ñ ##œ œWÞ"
ii) So now we know formulas for WW!" and . How can use these to we get a formula for ## # W# œ"#ÞÞÞ8 = ??? It's the same idea, but just a little more algebra.
For any 4 we know that Ð4 "Ñ$$ 4 œ $4 # $4 " . So we write this down substituting each value 4 œ "ß 4 œ #ß ÞÞÞß 4 œ 8
#"$$ œ$Ð"Ñ$Ð"Ñ" # $#$$ œ$Ð#Ñ$Ð#Ñ" # %$$$ œ$Ð$Ñ$Ð$Ñ" # ã Ð8"Ñ$$ 8 œ$Ð8Ñ$Ð8Ñ" # . Adding up the columns on both sides (with lots of cancellations on the left-hand side) gives ______Ð8"Ñ$### " œ$Ð" # ÞÞÞ8 Ñ$Ð"#ÞÞÞ8ÑÐ""ÞÞ"Ñ ÆÆÆ œ$ W#"! $ W W ß so
$# 8 $8 $8 " " œ $W#"! $W W . Then we solve to get W # .
$# 8 $8 $8 $W"! W œ $W # , so $# 8 $8 $8$W"! W Wœ# $ . We can substitute the formulas we already know for WW!" and and simplfy to ge
$# 8Ð8"Ñ 8 $8 $8 $[]# 8 8Ð8"ÑÐ#8"Ñ Wœ# $'œœœ...
Optional Material
If you know the binomial formula (from high school) and can therefore expand Ð4 "Ñ5 , then the same idea works for any natural number 55 . But the bigger is, the more lagebra is involved. An outline goes like this.
The formula for the “binomial coefficients” : 5 œÑ5x 66xÐ56Ñx
Suppose we have figured out formulas for WßWßWßÞÞÞßW!"# 5" . We know (from the binomial theorem) that for any 4 ,
Ð4"Ñ4œ5" 5"5" 4 5 5" 4 5" 5" 4ÞÞÞ" 5# "# $ Write this out for each value 4 œ "ß #ß ÞÞÞ8 .
25" " 5" œ5" " 5 5" " 5" 5" " 5# ÞÞÞ " "# $ $5" # 5" œ5" # 5 5" # 5" 5" # 5# ÞÞÞ " "# $ ÞÞÞÞÞÞ
Ð8 "Ñ5" 8 5" œ5" 8 5 5" 8 5" 5" 8 5# ÞÞÞ " . Add the columns: "# $ ______
Ð8"Ñ5" 1 œ5" Ð" 5 # 5 ÞÞÞ8 5 Ñ 5" Ð" 5" # 5" ÞÞÞ8 5" Ñ "#
5" Ð"5# # 5# ÞÞÞ 8 5" Ñ ÞÞÞ 8 $
œWWWÞÞÞW5" 5" 5" . "#55"5#! $ Then we solve for what we want:
WœÐ8"Ñ"W[]5" 5" 5" WÞÞÞWÎ 5" 55"5#!#$ " œÐ8"Ñ"WWÞÞÞWÎÐ5"Ñ[]5" 5" 5" #$5" 5# !
We are assuming that we already have formulas for WßWßÞÞÞWW5" 5# " , ! which we then substitute into this formula to get one closed, if complicated, formula for W8Þ5 in terms of Try it to find a formula for
%% % Wœ"#ÞÞÞ8œ% ...