# What is the limit of this series?

• Aug 3rd 2011, 03:05 AM
Supernova008
What is the limit of this series?
http://i632.photobucket.com/albums/u...i/formula1.png

where a, b and e are constants. g and n are obvious from the sum. How can I find the limit of this series as n approaches infinity? I already tried several approaches, but it's always problematic as the "g" is under the square root sign.
• Aug 3rd 2011, 04:33 AM
CaptainBlack
Re: What is the limit of this series?
Quote:

Originally Posted by Supernova008
http://i632.photobucket.com/albums/u...i/formula1.png

where a, b and e are constants. g and n are obvious from the sum. How can I find the limit of this series as n approaches infinity? I already tried several approaches, but it's always problematic as the "g" is under the square root sign.

First lump your constants, put $\displaystyle A=|ea|$, and $\displaystyle c=|b/ea|$ then you have:

$\displaystyle A \sum_{k=1}^n \frac{1}{n}\sqrt{(k/n)^2+c^2}$

The sum can be interpreted as a Riemann sum for an integral from 0 to 1 of a continuous function so the limit is:

$\displaystyle A \int_0^1 \sqrt{x^2+c^2}\; dx$

CB
• Aug 3rd 2011, 05:52 AM
Supernova008
Re: What is the limit of this series?
Quote:

Originally Posted by CaptainBlack
First lump your constants, put $\displaystyle A=|ea|$, and $\displaystyle c=|b/ea|$ then you have:

$\displaystyle A \sum_{k=1}^n \frac{1}{n}\sqrt{(k/n)^2+c^2}$

The sum can be interpreted as a Riemann sum for an integral from 0 to 1 of a continuous function so the limit is:

$\displaystyle A \int_0^1 \sqrt{x^2+c^2}\; dx$

CB

I am not a university student yet, so pardon my lack of such advanced mathematical knowledge. I am currently researching this riemann sum, but could you tell me why this is so?

EDIT: the first term of the sum is (1/n)*sqrt(1/n^2 + m^2)
however, the first term of the riemann sum is 1/n*sqrt(n^2+m^2), right? isn't that an inconsistency?
• Aug 3rd 2011, 06:30 AM
CaptainBlack
Re: What is the limit of this series?
Quote:

Originally Posted by Supernova008
I am not a university student yet, so pardon my lack of such advanced mathematical knowledge. I am currently researching this riemann sum, but could you tell me why this is so?

Quote:

EDIT: the first term of the sum is (1/n)*sqrt(1/n^2 + m^2)
however, the first term of the riemann sum is 1/n*sqrt(n^2+m^2) , right? isn't that an inconsistency?
What does the red text above refer to?

The partitioning points are $\displaystyle 1/n,\ 2/n,\ 3/n,\ ...\ ,\ n/n$ and $\displaystyle h=1/n$.

(not advanced mathematical knowlege, this was taught in what would now be years 11 or 12 when I was in secondary education)

CB
• Aug 3rd 2011, 06:45 AM
Supernova008
Re: What is the limit of this series?
Quote:

Originally Posted by CaptainBlack
What does the red text above refer to?

The partitioning points are $\displaystyle 1/n,\ 2/n,\ 3/n,\ ...\ ,\ n/n$ and $\displaystyle h=1/n$.

(not advanced mathematical knowlege, this was taught in what would now be years 11 or 12 when I was in secondary education)

CB

Yeah sorry, I got it already, thank you very much!