Lonely

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15 years, 72 days

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These are questions asked by Lonely

please help me here:

we have the following series:

1/2-Pi/(4*sqrt(3)) = sum(1/((6*n)^2-1), n = 1 .. Infinity)

 

(5-Pi)*(1/4) = sum((-1)^(n+1)/(2*n+1), n = 1 .. Infinity) {***}

 

-1/5+(1/6)*Pi-(1/12)*sqrt(3)*log((2+sqrt(3))/(2-sqrt(3))) = sum((-1)^n/(6*n+5), n = 1 .. Infinity)

 

please help me here:

i have :

h(x):=arctan(tan(x)*tan(x)):

how do i find:

int(h(x),x=0..Pi)

i think its value is pi^2/4

suppose if we also have

g(x):=arctan(cos(2 *x))+Pi/(4):

how do we find:

int(g(x)-h(x), x = 0 .. Pi)

thanks

 

 

please help me here:

we have a function:

f(x):=arctan(tan(x)*tan(x))

now let us look at the even derivative of this function at x = Pi/2. that is:

f^2(Pi/2) = -2

f^4(Pi/2) = -16

f^16(Pi/2) = -32

f^8(Pi/2) = 18944

f^10(Pi/2)= 1904128

is there a pattern in these numbers?

thanks

 

please help me here:

how can i integrate, say for example f(x) = cos(x), n times. means. how to find?

int(...int(int(int(cos(x),x),x),x)..,x)

thanks

 

please help me here:

i want to evaluate the integral:

int(exp(-xi^2*(3*n+xi)/(6*n^2+6*xi*n+xi^2)), xi = -infinity .. infinity)

thanks

 

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