Equation for a summed series with a limit

Hi,

I'm completely new to this. I am a software developer and haven't had the calculus books open for about a quarter of a century so I do aplologize if this is not in the correct forum. Further I'm not even sure if the question is phrased correctly.

I'm basically trying to figure out the value of the summed series

where there are x elements in the series as x gets large. d and r are constants.

Hoping you can help or point me in the right direction.

Cheers,

Colin.

Re: Equation for a summed series with a limit

I've taken my project to the next step using the equation you provided earlier and my next hurdle occurs with the addition of a second variable r2 and substituting the denominator of the original equation as the d variable in your solution. The latex probably explains better what I’m chasing;

Is this still a Riemann sum?

If I've got this right this should also converge to around 0.06 for d=4 and r = 1.

Re: Equation for a summed series with a limit

Quote:

Originally Posted by

**ColinJ** I've taken my project to the next step using the equation you provided earlier and my next hurdle occurs with the addition of a second variable r2 and substituting the denominator of the original equation as the d variable in your solution. The latex probably explains better what I’m chasing;

Is this still a Riemann sum?

If I've got this right this should also converge to around 0.06 for d=4 and r = 1.

Yes it is still a Riemann sum, and the limit is equal to the integral

That is a far more elaborate integral than the one in the previous problem, and I doubt whether it is possible to give the answer in a closed form. For particular values of d, r_1 and r_2, you can presumably get Mathematica or Wolfram to churn out an answer.

Re: Equation for a summed series with a limit

Hi,

I've spent a fair amount of time on my project and discovered that I need another term in the denominator (as below).

I'm guessing this becomes the integral;

.

But, I can't for the life of me solve this thing. Any help is appreciated once again. I'm fairly sure it converges to around 0.0412468 for d = 5 and r = 1.

Regards,

Colin.

Re: Equation for a summed series with a limit

Quote:

Originally Posted by

**ColinJ** Hi,

I've spent a fair amount of time on my project and discovered that I need another term in the denominator (as below).

I'm guessing this becomes the integral;

.

But, I can't for the life of me solve this thing. Any help is appreciated once again. I'm fairly sure it converges to around 0.0412468 for d = 5 and r = 1.

You are right about the integral. As it happens, this complicated-looking integral can be evaluated fairly easily by using complex numbers and the method of contour integration. If you make the substitution then the integral becomes

(**)

where the integral is taken round the unit circle C. If then Cauchy's integral formula gives the answer as (I have skipped all the details of the calculation). If d=5 and r=1, that becomes fairly close to the answer that you wanted.

(**) **Edit. **I have just seen that this integral looks confusing. The "dz" in the numerator is just the usual notation for an integral with respect to z, but in the denominator "d" is a constant and " " means d times z squared.

Re: Equation for a summed series with a limit

Thanks once again.

My fault on that confusing integral. Obviously d is an inappropriate variable where integrals are involved. And, I have some reading to do.

Much appreciated.

Re: Equation for a summed series with a limit

would you mind posting some of the details of the calculation? i cant follow (at least the algebra) in converting from to z.

i got to a point like this and gave up:

after making the substitution.

though i think i may be hijacking this thread.. if i am please correct me.

Quote:

Originally Posted by

**Opalg** You are right about the integral. As it happens, this complicated-looking integral can be evaluated fairly easily by using complex numbers and the method of

contour integration. If you make the substitution

then the integral becomes

(**)

where the integral is taken round the unit circle C. If

then Cauchy's integral formula gives the answer as

(I have skipped all the details of the calculation). If d=5 and r=1, that becomes

fairly close to the answer that you wanted.

(**)

**Edit. **I have just seen that this integral looks confusing. The "dz" in the numerator is just the usual notation for an integral with respect to z, but in the denominator "d" is a constant and "

" means d times z squared.

Re: Equation for a summed series with a limit