Euler's formula

[Nazzhoul]

Yes, this is for math class.

"Fact" 1:

sin(t) = t - [(t^3)/3!] + [(t^5)/5!] - [(t^7)/7!] ...
In English, Sine of t EQUALS (t) MINUS (t cubed over 3 factorial) PLUS (t to the 5th over 5 factorial) MINUS (t to the seventh over seven factorial), and this continues in an infinite series.

"Fact" 2:

cos(t) = 1 - [(t^2)/2!] + [(t^4)/4!] - [(t^6)/6!] ...

"Fact" 3:

e^t = 1 + t + [(t^2)/2!] + [(t^3)/3!] + [(t^4)/4!] ...
e being Euler's number.



"Statement" 1:

i * sin(t) = it - [(i{t^3})/3!] + [(i{t^5})/5!] - [(i{t^7})/7!] ...
(i in this problem is the irrational number i, the square root of -1.)

"Statement" 2:

e^it = 1 + it - [(t^2)/2!] - [(i{t^3})/3!] + [(t^4)/4!] + [(i{t^5}/5!] ...

"Statement" 3:

i * sin(t) + cos(t) = e^it

"Statement" 4:

e^i(pi) = -1
And therefore,
1 + e^i(pi) = 0




"Question"

Assume the "facts" are true. The "statements" are an inductive proof of Euler's formula. However, the reasoning of the statements is WRONG somewhere, and we are supposed to "object" to something about it. Your mission, should you choose to accept it, is to find out what the objection is supposed to be.

Thanks in advance.

[Ash]

found it lol but im not gonna tell u

[Nazzhoul]

found it lol but im not gonna tell u

y not ash
it may just be a typo im not sure

[Cryptonic]

do your own homework lol

[Dean Newton]

The formulas in the OP don't properly deal with raising i to exponents. i^2 = -1, i^3 = -i, etc.

[vornksr]

the formatting is so ugly i don't really want to read it. do these forums not have tex support?

I don't really understand the nature of the question. i isn't irrational, however.

[Glip]

the formatting is so ugly i don't really want to read it. do these forums not have tex support?

I don't really understand the nature of the question. i isn't irrational, however.

I assume you mean LaTeX, but yes, these forums DO support it. The OP just didn't use it.

The OP is saying that there is something wrong with one of the four "statements" and is asking us to determine what it is.

[vornksr]

well i was being lazy and not making a distinction, but sure, I mean latex.

it looks like a pretty standard proof of euler's formula. it doesn't justify rearranging the terms between statements 2 & 3, which is pretty important, but i'm not sure that a high-school level class (which i assume this is) would even talk about issues like absolute convergence.

[Nazzhoul]

The formulas in the OP don't properly deal with raising i to exponents. i^2 = -1, i^3 = -i, etc.

Only place this is relevant is in e^it, where that IS accounted for. Coefficients are, in order, i^0, i^1, i^2 etc etc.

I assume you mean LaTeX, but yes, these forums DO support it. The OP just didn't use it.

The OP is saying that there is something wrong with one of the four "statements" and is asking us to determine what it is.

I dun no how 2 use latex

[Nazzhoul]

well i was being lazy and not making a distinction, but sure, I mean latex.

it looks like a pretty standard proof of euler's formula. it doesn't justify rearranging the terms between statements 2 & 3, which is pretty important, but i'm not sure that a high-school level class (which i assume this is) would even talk about issues like absolute convergence.

Not sure what you're saying here. Going from 2 to 3, I add cos t and sin t and that yields the same infinite series as e^it. Is that something that shouldn't happen?

[vornksr]

When you add the series expressions for cos(t) and isin(t) you rearrange the order of the terms by interleaving them. (You have 1 from cos then t^1 from sin then t^2 from cos, etc. rather than 1 - t^2 + t^4 ... + t - t^3... You see what I mean?) Technically, you can't always rearrange the order of terms in a series and be guaranteed that the series will converge to the same number. It turns out that you can when it comes to cos, sin, and exp, but the proof doesn't justify that step.

[clementine]

When you add the series expressions for cos(t) and isin(t) you rearrange the order of the terms by interleaving them. (You have 1 from cos then t^1 from sin then t^2 from cos, etc. rather than 1 - t^2 + t^4 ... + t - t^3... You see what I mean?) Technically, you can't always rearrange the order of terms in a series and be guaranteed that the series will converge to the same number. It turns out that you can when it comes to cos, sin, and exp, but the proof doesn't justify that step.

vorn, where were you when I needed help with my complex analysis problem sets!?

[vornksr]

hm clearly i was lurking, which is probably a sign i'm scum