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edited Jan 28 at 15:20

I have a question regarding transforming a summation to an integral using the Euler–Maclaurin formula. Imagine I have this sum $$\sum_{i=0}^{m} f(i) \qquad \text{with} \qquad f(i)= \exp [(\frac{a}{b+c+i})^2] $$ where the upper limit $m$ is a finite non-negative integer, and $a,b,c\in\mathbb{R}$. My major is not Math; searching on the internet, I see that I can use the Euler–Maclaurin formula as follows (the source of the picture is WIKIPEDIAWIKIPEDIA ) where it is written that for p a positive integer and a function f(x) that is p times continuously differentiable on the interval [m,n]

My questions:

Can I use this formula for my function $f(i)$? More precisely, I mean does this part a function f(x) that is p times continuously differentiable on the interval [m,n] apply to my case too?
Then, will the new summation (the 3rd term in the formula) be convergent?
Is the error term $R_p$ negligible compared to the other terms?
Is there any simpler formula that I can use to transform my summation to an integral?

I appreciate any help and comments.

I have a question regarding transforming a summation to an integral using the Euler–Maclaurin formula. Imagine I have this sum $$\sum_{i=0}^{m} f(i) \qquad \text{with} \qquad f(i)= \exp [(\frac{a}{b+c+i})^2] $$ where the upper limit $m$ is a finite non-negative integer, and $a,b,c\in\mathbb{R}$. My major is not Math; searching on the internet, I see that I can use the Euler–Maclaurin formula as follows (the source of the picture is WIKIPEDIA) where it is written that for p a positive integer and a function f(x) that is p times continuously differentiable on the interval [m,n]

My questions:

Can I use this formula for my function $f(i)$? More precisely, I mean does this part a function f(x) that is p times continuously differentiable on the interval [m,n] apply to my case too?
Then, will the new summation (the 3rd term in the formula) be convergent?
Is the error term $R_p$ negligible compared to the other terms?
Is there any simpler formula that I can use to transform my summation to an integral?

I appreciate any help and comments.

I have a question regarding transforming a summation to an integral using the Euler–Maclaurin formula. Imagine I have this sum $$\sum_{i=0}^{m} f(i) \qquad \text{with} \qquad f(i)= \exp [(\frac{a}{b+c+i})^2] $$ where the upper limit $m$ is a finite non-negative integer, and $a,b,c\in\mathbb{R}$. My major is not Math; searching on the internet, I see that I can use the Euler–Maclaurin formula as follows (the source of the picture is WIKIPEDIA ) where it is written that for p a positive integer and a function f(x) that is p times continuously differentiable on the interval [m,n]

My questions:

Can I use this formula for my function $f(i)$? More precisely, I mean does this part a function f(x) that is p times continuously differentiable on the interval [m,n] apply to my case too?
Then, will the new summation (the 3rd term in the formula) be convergent?
Is the error term $R_p$ negligible compared to the other terms?
Is there any simpler formula that I can use to transform my summation to an integral?

I appreciate any help and comments.

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asked Jan 28 at 15:02

MsMath

Question on transforming a sum to an integral using the Euler–Maclaurin formula.

I have a question regarding transforming a summation to an integral using the Euler–Maclaurin formula. Imagine I have this sum $$\sum_{i=0}^{m} f(i) \qquad \text{with} \qquad f(i)= \exp [(\frac{a}{b+c+i})^2] $$ where the upper limit $m$ is a finite non-negative integer, and $a,b,c\in\mathbb{R}$. My major is not Math; searching on the internet, I see that I can use the Euler–Maclaurin formula as follows (the source of the picture is WIKIPEDIA) where it is written that for p a positive integer and a function f(x) that is p times continuously differentiable on the interval [m,n]

My questions:

Can I use this formula for my function $f(i)$? More precisely, I mean does this part a function f(x) that is p times continuously differentiable on the interval [m,n] apply to my case too?
Then, will the new summation (the 3rd term in the formula) be convergent?
Is the error term $R_p$ negligible compared to the other terms?
Is there any simpler formula that I can use to transform my summation to an integral?

I appreciate any help and comments.