math question

[eman7613]

So - i was doing sum comparison tests between my library and my calculator - and i was doing summations. and i accidentally did the sum of x^3, from x = 1 to x = -5. I expected an error (how can you count up from 1 to -5 right?) but instead i got the result of 100. So I did some testing: on my TI-89 and TI-nspireCAS (both computer algebra systems) i got the result of 100. On my TI-84 and Casio (regular calculators) I got an error, and in Maxima (another computer algebra system) i got 0.


Does anyone know how to get that result? or is this some strangeness that TI has simply added to their calculators?

[Kram1032]

TI-92 solves it like that:

sum(x³,x,a,b)=
-a^4/4 +a³/2 -a²/4 +b²(b+1)²/4

for a=1 and b=-5, you get 100

that sum simply works because TI is able to solve the general case

[suvakas]

What do you eat Kram?
What makes you know all this stuff?

[benn]

You are a machine Kram.

[PureSpider]

Thats simple integration rules if I understand that right... nothing too complicated

[suvakas]

IF

[eman7613]

TI-92 solves it like that:

sum(x³,x,a,b)=
-a^4/4 +a³/2 -a²/4 +b²(b+1)²/4

for a=1 and b=-5, you get 100

that sum simply works because TI is able to solve the general case

I did some more work, and found out most of it is different conventions - but TI' s formula is actually different, that case was just coincidence!

for: sum(x³,x,a,b)
if a > b > 0 , then it performs sum(x³,x,(a-1)*-1,b)
if a > 0 > b, then is does sum(x³,x,a,|b-1|)
if 0 > a > b, the result is sum(x³,x,a,|b+1|)

and the expansion of x^3 that it gives dose not work the same as the just stated algorithum for most input

took me a little to find out the pattern

Im just going to make it throw an error

[Kram1032]

why would you do that?
If the general case can be solved, you can even solve it for floats and complex numbers....

@"How do you do":
simply type:

Code: Select all

2nd 4 x ^ 3 , x , a , b ) ENTER

gives me, after putting it on one ratio,

Code: Select all

-(a+b)*(a-b-1)*(a²-+b*(b+1))
____________________________
              4

then I say:

Code: Select all

2nd (-) 2nd k a=1 and b=-5

and my TI returns

Code: Select all

100

That's all the magic. I love it^^
We did NOT learn (yet, if we ever will), how to solve such sums generally. It's just what the TI tells me.

As said, interestingly, it's even possible to "count" from, say, -φ to π*i.... that gives:

Code: Select all

4*π^4-4*π²-(√5+3)*(3*√5+7)   π³
__________________________-_____ * i
           16                2

or as floats:

Code: Select all

17.3988036807-15.5031383402*i

[Camox]

Kram, you're also randomly janitor at your school ?

http://www.youtube.com/watch?v=Sjd6n-kyasI

[Kram1032]

nope...
gotta think about the real result though... hum... 3? 5?
Way too high maths to me... gotta go back to generalise spheres to any dimension. Not so much headache^^

(Btw: Ever tried to PROOF that 1+1=2?)

[Camox]

[eman7613]

Again, that simply a misnomer because of the equation I selected.

Try to do the summation on x*ln(x), it is a summation that can not be generalized, and you will find the ti can not sum it if you specify either of the limits as complex, you will not get a result.

However, it also disproves my previous algorithm since you cant take the log of a negative number and the TI still gives a result


back to square one!!

(btw, yes i have! other fun ones, proving 5^(1/2) is irrational, and deriving the direct formula for the nth Fibonacci number are some of my favorites!)

[Kram1032]

n^(1/k) (n,k= positive integers) is proofen via contradictions, right?

you try to proof, that n^(1/k)=a/b with a,b being positive ints again and somehow you get a contradiction or a limitation (if you assume n=c^k with c again being a positive integer, it'll work as n^(1/k)=c/1 but else, you simply can't proof, that n^(1/k) can be written as a/b, due to a contradiction, which automatically tells you the opposite...)

I read a poem or two which proofed that 2^(1/2) is irrational xD

[eman7613]

n^(1/k) (n,k= positive integers) is proofen via contradictions, right?

you try to proof, that n^(1/k)=a/b with a,b being positive ints again and somehow you get a contradiction or a limitation (if you assume n=c^k with c again being a positive integer, it'll work as n^(1/k)=c/1 but else, you simply can't proof, that n^(1/k) can be written as a/b, due to a contradiction, which automatically tells you the opposite...)

I read a poem or two which proofed that 2^(1/2) is irrational xD

the easy one is proof via contradiction yes - but the most important part is that if n^(1/l)=a/b, that a/b is an irreducible fraction - although if you dont you eventually conclude that the fraction is infinitely reducible , which is also patently absurd and satisfies proof by contradiction

[Kram1032]

ah, yes, forgot that point^^