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Thanks to aamath.
Input: f(x+h) = f(x) + h*f'(x) + h^2/2*f''(x) + O(h^3)
Output:
2
h / 3\
f(x + h) = f(x) + h f'(x) + -- f''(x) + O\h /
2
Input: h = -((f'(x)/f''(x))*(1 - sqrt(1 - (2*f(x)*f''(x))/f'(x)^2)))
Output:
/ _________________\
| / 2 f(x) f''(x)|
f'(x) | / 1 - -------------|
h = - ------ |1 - / 2 |
f''(x) \ \/ f'(x) /
Input: sqrt(42)/z=root((1+1/(1+1/(x^2+1/b)))^3,6)/(3^d/(5-e + 42/(3 + f))+sqrt((2/(1-1/(1+1/7))))+sqrt(1/(2+3)+3)^(sqrt(21/(38-w))))
Output:
6_________________
/ 3
/ / 1 \
/ |1 + ----------|
/ | 1 |
/ | 1 + ------|
/ | 2 1|
__ / | x + -|
\/42 \/ \ b/
---- = -------------------------------------------------------------
z ______
/ 21
_________ / ------
/ 2 \/ 38 - w
d / --------- / _________\
3 / 1 | / 1 |
------------- + / 1 - ----- + | / ----- + 3|
42 / 1 \\/ 2 + 3 /
5 - e + ----- / 1 + -
3 + f \/ 7
Input: int(int(int(psi^2, x = -inf .. inf), y = -inf .. inf), z = -inf .. inf) = 1
Output:
oo oo oo
/ / /
| | | 2
| | | psi dx dy dz = 1
| | |
/ / /
-oo -oo -oo
Input: A_TR = x*sqrt(x^2-1)/2 - int(sqrt(t^2-1), t = 1 .. x)
Output:
x
______ /
/ 2 | ______
x \/ x - 1 | / 2
A = ----------- - | \/ t - 1 dt
TR 2 |
|
/
1
Input: sqrt(e) = 1+1/(1+1/(1+1/(1+1/(5+1/(1+1/(1+1/(9+1/(1+1/(1+...)))))))))
Output:
_ 1
\/e = 1 + ---------------------------------------
1
1 + -----------------------------------
1
1 + -------------------------------
1
1 + ---------------------------
1
5 + -----------------------
1
1 + -------------------
1
1 + ---------------
1
9 + -----------
1
1 + -------
1 + ...
Input: e^x = 1 + x + x^2/2! + x^3/3! + x^4/4! + ... = 1 + sum(x^n/n!, n = 1 .. inf)
Output:
oo
2 3 4 ===== n
x x x x \ x
e = 1 + x + -- + -- + -- + ... = 1 + > --
2! 3! 4! / n!
=====
n = 1
Input: (1/4)*pi*sqrt(2) = sum((-1)^(k+1)/(4*k + 1) + (-1)^(k+1)/(4*k - 3), k = 1 .. inf) = 1 + 1/3 - 1/5 - 1/7 + 1/9 + 1/11 - ...
Output:
oo
===== / k + 1 k + 1\
1 __ _ \ |(-1) (-1) | 1 1 1 1 1
- || \/2 = > |--------- + ---------| = 1 + - - - - - + - + -- - ...
4 / \ 4 k + 1 4 k - 3 / 3 5 7 9 11
=====
k = 1
Input: 2/pi=sqrt(1/2)*sqrt(1/2+1/2*sqrt(1/2))*sqrt(1/2+1/2*sqrt(1/2+1/2*sqrt(1/2)))*...
Output:
___________ ______________________
_ / _ / ___________
2 /1 / 1 1 /1 / / _
-- = / - / - + - / - / 1 1 / 1 1 /1 ...
__ \/ 2 \/ 2 2 \/ 2 / - + - / - + - / -
|| \/ 2 2 \/ 2 2 \/ 2
Input: pi = 3/4*sqrt(3) + 24*int(sqrt(x - x^2), x = 0 .. 1/4) = (3*sqrt(3))/4 + 24 * (1/12 - 1/(5*2^5) - 1/(28*2^7) - ...)
Output:
1
-
4
/
| ______ _
__ 3 _ | / 2 3 \/3 / 1 1 1 \
|| = - \/3 + 24 | \/ x - x dx = ----- + 24 |-- - ---- - ----- - ...|
4 | 4 |12 5 7 |
| \ 5 2 28 2 /
/
0
Input: int(z^2, z = 1 .. root(3, 3)) * cos((3*pi)/9) = ln(root(e, 3))
Output:
3_
\/3
/ __
| 2 3 || / 3_\
| z dz cos ---- = ln\\/e/
| 9
/
1
Input: x\ = (x_1 + x_2 + x_3 + ... + x_n)/n = (1/n)*sum(x_i, i = 1 .. n)
Output:
n
x + x + x + ... + x =====
_ 1 2 3 n 1 \
x = ----------------------- = - > x
n n / i
=====
i = 1
Input: zeta(s) = (1 / (1 - (1/2^s))) * (1 / (1 - (1/3^s))) * (1 / (1 - (1/5^s))) * (1 / (1 - (1/7^s))) * ... = prod(1 / (1 - (1/p^s)), p_prime)
Output:
=====
1 1 1 1 | | 1
zeta(s) = ------ ------ ------ ------ ... = | | ------
1 1 1 1 | | 1
1 - -- 1 - -- 1 - -- 1 - -- | | 1 - --
s s s s p s
2 3 5 7 prime p
Input: int((x^2+a)/b,x) = (1/b)*int(x^2+a,x) = (1/b)*(x^3/3 + a*x) + C
Output:
/
| 2 / / 3 \
| x + a 1 | / 2 \ 1 |x |
| ------ dx = - | \x + a/ dx = - |-- + a x| + C
| b b | b \ 3 /
| /
/
Input: sin(a)/a = cos(a/2) * cos(a/4) * cos(a/8) * cos(a/16) * ... = prod(cos(a/2^n), n = 1 .. inf)
Output:
oo
=====
sin a a a a a | | a
----- = cos - cos - cos - cos -- ... = | | cos --
a 2 4 8 16 | | n
| | 2
n = 1
Input: A_T = [sqrt(a/b), 0, 0; 0, sqrt(a/b), 0; 0, 0, sqrt(a/b)]^-1
Output:
-1
/ _ \
| /a |
| / - 0 0 |
| \/ b |
| |
| _ |
| /a |
A = | 0 / - 0 |
T | \/ b |
| |
| _ |
| /a |
| 0 0 / - |
\ \/ b /
Input: lim(1/x^2 - (cos(x)/x)^2, x -> inf) = 1
Output:
/ 2\
| 1 /cos x\ |
lim |-- - |-----| | = 1
| 2 \ x / |
x -> oo \x /