Index and bound shift in converting a sum into integral

In summary, the conversation discusses an equality and its meaning in relation to integrals and sums. The expression presented is considered to be meaningless without additional context. The concept of index shifting is also discussed, as well as the differences between integrals and sums. The conversation concludes with an explanation of how the inequalities can be reversed if the function is increasing.
  • #1
7
1
Considering the below equality (or equivalency), could someone please explain how the bounds and indices are shifted?
$$\sum_{i=2}^{k}(h_i/f_{i-1})=\int_{1}^{k}(h(i)/f(i))di$$
 
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  • #2
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.
 
  • #3
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.
 
  • #4
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.)
 
  • #5
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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  • #6
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.
 
  • #7
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.
 

1. What is the purpose of converting a sum into an integral?

Converting a sum into an integral allows us to approximate the area under a curve by breaking it into smaller, more manageable sections. This is useful in many scientific and mathematical applications, such as calculating the distance traveled by an object with varying velocity or determining the volume of an irregularly shaped object.

2. How does the index and bound shift method work?

The index and bound shift method involves replacing the summation index with a variable, usually represented by the letter "x". The bounds of the summation are then converted into limits of integration. This allows us to rewrite the sum as an integral, which can then be solved using integration techniques.

3. Can any sum be converted into an integral using the index and bound shift method?

No, not all sums can be converted into integrals using this method. The sum must have a finite number of terms and the terms must be continuous functions. Additionally, the sum must be able to be expressed as a definite integral with a finite interval of integration.

4. What are the benefits of using the index and bound shift method?

Converting a sum into an integral using the index and bound shift method allows for easier computation and more accurate results. It also allows us to apply integration techniques, such as the fundamental theorem of calculus, to solve problems that would be difficult or impossible to solve using summation methods.

5. Are there any limitations to using the index and bound shift method?

One limitation of the index and bound shift method is that it can only be applied to sums with a finite number of terms. Additionally, it may not be the most efficient method for solving certain problems, as it requires knowledge of integration techniques. In some cases, other methods such as the trapezoidal rule or Simpson's rule may be more suitable for approximating the area under a curve.

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