Answer :
This is a Riemann Sum.
First of all, Δx = 1/n.
So, now we have
lim (n→∞) Δx [1/(1+Δx) + 1/(1+2Δx)+ ... + 1/(1+nΔx)]
= lim (n→∞) Σ(k = 1 to n) [1/(1 + kΔx)] Δx.
Since x <---> a + kΔx
So, if we take a = 0, then b = 1, so that Δx = (b - a)/n = 1/n, etc.
(the choice of a is arbitrary, along as you choose b appropriately so that b - a = 1 in this case).
However, due to the (1 + kΔx) combination, I'll set a = 1 so that b = 2.
Then, f(1 + kΔx) <-----> f(x) ==> f(x) = 1/x.
Hence, this sum represents the integral
∫(x = 1 to 2) (1/x) dx, choice d.
First of all, Δx = 1/n.
So, now we have
lim (n→∞) Δx [1/(1+Δx) + 1/(1+2Δx)+ ... + 1/(1+nΔx)]
= lim (n→∞) Σ(k = 1 to n) [1/(1 + kΔx)] Δx.
Since x <---> a + kΔx
So, if we take a = 0, then b = 1, so that Δx = (b - a)/n = 1/n, etc.
(the choice of a is arbitrary, along as you choose b appropriately so that b - a = 1 in this case).
However, due to the (1 + kΔx) combination, I'll set a = 1 so that b = 2.
Then, f(1 + kΔx) <-----> f(x) ==> f(x) = 1/x.
Hence, this sum represents the integral
∫(x = 1 to 2) (1/x) dx, choice d.
If n is a positive integer, then lim n→∞ 1/n[(1/n)² + (2/n)² +...+((n-1)/n)²] can be expressed as; the integral from 1 to 2 of 1/x dx
- The relationship we are trying to derive would be gotten from expressing the integral in terms of what we call Riemann sums.
The procedure is to Specifically divide the intervals from 1 to 2 into a total of n subintervals with equal lengths and then rectangles would be constructed as indicated in the attached diagram.
- From the attached diagram, we can see that;
The integral will be gotten by addition of the areas of the rectangles after which we will now take a limit as n approaches ∞.
For example the first rectangle to the left of the attached diagram has a base length; 1 + ¹/ₙ ₋ ₁ = ¹/ₙ
height; f(1 + ¹/ₙ) = 1/(1 + ¹/ₙ)
This means it will have an area of;
A = (¹/ₙ)(1/(1 + ¹/ₙ)).
- Now, If we repeat the same process of the area above with each of the rectangles and sum them as well as taking the limit as n approaches infinity, we will arrive at the expression in the question.
Thus, it means that the given expression can be expressed as the integral from 1 to 2 of 1/x dx
Read more on Riemann sums at; https://brainly.com/question/84388
