I have been doing infinite series in Calc. We have been learning all the tests for convergence and divergence but not really knowing what the sum is. Our professor said that there is no real trick in finding the sum unless its a geometric series.
So I went on Maple and put in some series that I know converged.
sry i dont really know latex
sum(1/n^3, n = 1 .. infinity)
for that particular one i got the 3*zeta. In the past I also got (I dont remember the equations) answers with Psi, psi, and even imaginary numbers. (I think the imaginary was an approximation though.
So i have two questions. What do Psi and zeta mean in mathematical terms? I looked up Psi on wikipedia, but they just defined it using the integral, series, and some other ways. They didn't actually tell me what it was used for.
My other question is why do they appear in my summation answer? I especially dont understand why a (i) would appear in a decimal approximation of a summation.
this equation gave me a whole gamut of greek sybmols and imaginary numbers
the answer i got in maple was
-1/3+1/3*Psi(1/2+1/2*I*sqrt(3))-1/6*I*Pi*tanh(1/2*Pi*sqrt(3))+1/6*sqrt(3)*Pi*tanh(1/2*Pi*sqrt(3))+1/3*gamma
the answer is pretty long, but i suppose that most of you have some sort of symbolic math program that you can use to find this. The (I) is in the second and third terms. When I approximate the answer to decimal the imaginary part of the complex number is only .00003(I). Still you shouldnt get an imaginary number.
I also noticed that when i find the sum of the first 999 terms i get a huge fraction without greek symbols of (I), but when I add the first 1000 terms ( just one more term mind you) i get an answer with greek symbols and (i). when i list the first 1000 terms i dont get any greek symbols or imaginary numbers. could it just be a weird maple thing?