# Thread: Infinite Series in Maple

1. ## Infinite Series in Maple

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.

2. Originally Posted by redier
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.
Since the reals are closed under the operations of addition and multiplication
$\displaystyle \bold{i}$ is not going to appear as the sum of a series using just these
operations.

Can you tell us which series gave a complex sum?

RonL

3. Originally Posted by redier
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.
Its likely what it gave was:

$\displaystyle \zeta(3)=\sum_1^{\infty} 1/n^3$

as this is a special case of the Riemann zeta function:

$\displaystyle \zeta(z)=\sum_1^{\infty} 1/n^z$

You can find out more about the zeta function here, and in numerous popular
books about it.

RonL

4. this equation gave me a whole gamut of greek sybmols and imaginary numbers
$\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^3+1}$

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?

5. Originally Posted by redier
this equation gave me a whole gamut of greek sybmols and imaginary numbers
$\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^3+1}$

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?
I guess that the imaginaries actualy cancel out (they must do if M.
is right as the sum must be real).

RonL