Index and bound shift in converting a sum into integral

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Discussion Overview

The discussion revolves around the conversion of a sum into an integral, specifically examining the shifting of bounds and indices in the expression $$\sum_{i=2}^{k}(h_i/f_{i-1})=\int_{1}^{k}(h(i)/f(i))di$$. Participants explore the validity and implications of this equivalency, addressing both theoretical and practical aspects.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant questions the meaningfulness of the expression, suggesting that the index in the sum and the number of elements cannot be directly carried over to the integral form as presented.
  • Another participant agrees with the previous point, noting that the expression appears in literature but lacks clarity without additional context regarding the nature of the functions involved.
  • Concerns are raised about whether ##f## is a function of a real variable or a sequence, highlighting the need for clarification on definitions.
  • A participant introduces inequalities related to the integral test for convergence, indicating that while the sum and integral are related, they do not equate directly, and the index remains unchanged.
  • There is a discussion about the nature of the variable ##i##, with participants noting that in integrals, ##i## is treated as a real or complex number, whereas in summations, it is an integer.
  • Clarification is sought regarding the type of integral being referenced, with mentions of Riemann integrals and Stieltjes integrals, suggesting that the expression may require additional complexity if the latter is intended.

Areas of Agreement / Disagreement

Participants express disagreement regarding the validity of the original expression and its interpretation. There is no consensus on how to properly shift bounds and indices, and the discussion remains unresolved.

Contextual Notes

Participants highlight limitations in the original expression due to missing context and assumptions about the nature of the functions involved. The discussion also reflects uncertainty regarding the definitions of the variables and the types of integrals being considered.

Alex_F
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Considering the below equality (or equivalency), could someone please explain how the bounds and indices are shifted?
$$\sum_{i=2}^{k}(h_i/f_{i-1})=\int_{1}^{k}(h(i)/f(i))di$$
 
Last edited:
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As it stands the expression is meaningless. You can approximate an integral with a sum, but the index in the sum and the number of elements in the sum cannot be carried over in the way you present it.
 
Svein said:
As it stands the expression is meaningless. You can approximate an integral with a sum, but the index in the sum and the number of elements in the sum cannot be carried over in the way you present it.
Actually this is not written by me. I have seen this in a book and also couple of articles. So I suppose that is correct.
 
I agree with @Svein -- without some additional context, it's difficult to understand what the parts of the equation is supposed to represent. For example, is ##f## a function of some real variable, or is it a sequence? (However, a sequence is generally considered to be a function defined on some set of integers.)
 
Generally what it holds, given that the function ##g(i)## is monotonically decreasing is that $$\int_N^{M+1}g(i)di\leq\sum_{i=N}^Mg(i)\leq g(N)+\int_N^{M}g(i)di$$

So as you can see we don't have equality but a pair of inequalities that "sandwiches" the sum, we don't have index shifting (##i## remains ##i## everywhere), but the bounds are kind of shifted.
For more details check
https://en.wikipedia.org/wiki/Integral_test_for_convergence#Proof

if g is increasing then the inequalities are reversed that is it holds:
$$\int_N^{M+1}g(i)di\geq\sum_{i=N}^Mg(i)\geq g(N)+\int_N^{M}g(i)di$$
 
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Delta2 said:
(i remains i everywhere),
In the integrals, i must be a real or complex number (if you mean a Stieltjes integral your expression must be a bit more complicated). In the summation, i is an integer.
 
Svein said:
In the integrals, i must be a real or complex number (if you mean a Stieltjes integral your expression must be a bit more complicated). In the summation, i is an integer.
No those are Riemann integrals. I guess the symbol i is a bit overloaded there.
 

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