Evaluate Summation: \sum_{i=1}^{50}\frac{1}{(100-i)^{1/2}}

[sara_87]

Homework Statement

How do i evaluate the following sum

$$\sum_{i=1}^{50}\frac{1}{(100-i)^{1/2}}$$

Homework Equations

The Attempt at a Solution

i haven't a clue on how to do this, can someone please give me a hint?
thank you

 

[LCKurtz]

I think you don't, at least not in terms of a nice closed formula. You can evaluate it with something like Maple or approximate it with an integral. What is the background for where this problem came from?

 

[sara_87]

thanks,
it's just a part of a numerical problem.
I used MATLAB to find the sum :)

but, if in the denominator, i had:
100a-i

where a is constant, how would i evaluate that? is there a formula?
(matlab surely wouldn't work here).

 

[Lancelot59]

I think you don't, at least not in terms of a nice closed formula. You can evaluate it with something like Maple or approximate it with an integral. What is the background for where this problem came from?

I agree. There isn't any formula I know of that will solve that, and I can't see any way of re-arranging it into a suitable form.

 

[Tedjn]

If a is a constant, write your Matlab function to take it in as a parameter.

 

[sara_87]

thanks both.

so, ur saying that if i evaluate it in MATLAB then it should give me an answer in terms of a?

 

[Count Iblis]

You can also use the Euler-Maclaurin sum formula.

 

[vaibhav1803]

You can also use the Euler-Maclaurin sum formula.

Euler maclaurin won't fit easily here, its for sums that can be approximated to integrals, if I'm not wrong,
here the domain of the sum is really small, plus due to the square root makes it worse (see graph of 1/sqrt(x), it apparently ceases to fall/falls very slowly after a certian interval)

hence i suggest use of recursive algorithms to find it, the only way to get the accurate sum from a mathematical standpoint.

 

[Lancelot59]

Or you could just add up all the terms.

 

[Count Iblis]

Euler maclaurin won't fit easily here, its for sums that can be approximated to integrals, if I'm not wrong,
here the domain of the sum is really small, plus due to the square root makes it worse (see graph of 1/sqrt(x), it apparently ceases to fall/falls very slowly after a certian interval)

hence i suggest use of recursive algorithms to find it, the only way to get the accurate sum from a mathematical standpoint.

Euler-Maclaurin works fine in this case. As long as the start and end points at which the summand is evaluated are not close to singularities, you'll get an asymptotic series that converges to quite close to the answer before it starts to diverge.

 

[Count Iblis]

Answer obtained by adding up the numbers on my antique HP-28s calculator

5.97865123618

Answer obtained by Euler-Maclaurin summation, also using my HP-28s calculator

5.97865123616

The difference is only due to roundoff errors. I tried to minimize that by starting with the smallest terms and adding the largest terms at the end. For the Euler-MacLaurin summation, I only needed to go as far as the B_4/4! f^(3) term. I did include the next term, but that did not contribute to the result.

 

[sara_87]

thank you very much,
this helps a lot
:)