(1/1+(4/n))(4/n)+(1/1+(8/n))(4/n)+(1/1+(12/n))(4/n)+....+(1/1+(4n/n))(4/n)
is a right Riemann Sum for a certain definite integral.
Integral 1 to b, f(x) dx.
using a partition of the interval [1,b] into n subintervals of equal length.
Then the upper limit of integration must be: b=?
and the integrand must be function f(x)=?
To determine the upper limit of integration (b) and the integrand (f(x)), we need to analyze the expression you have provided, which is a right Riemann sum for the definite integral.
A right Riemann sum is given by the formula: Σ [f(xi) * Δx], where Σ represents the summation notation, f(xi) is the value of the function evaluated at the right endpoint of each subinterval, and Δx represents the width of each subinterval.
In your expression, the terms (1/1+(4/n))(4/n), (1/1+(8/n))(4/n), (1/1+(12/n))(4/n), ..., (1/1+(4n/n))(4/n) represent the function values evaluated at the right endpoints of the subintervals. We can observe that each term is of the form (1/1+(4n/n))(4/n), which simplifies to (1/1+4)(4/n). Thus, the value of the function f(x) is 4/(1+4) = 4/5.
The subintervals of your expression have equal length, so we can express the width of each subinterval, Δx, as (b - 1)/n, where n is the number of subintervals.
Now, let's analyze the expression further:
(1/1+(4/n))(4/n) + (1/1+(8/n))(4/n) + (1/1+(12/n))(4/n) + ... + (1/1+(4n/n))(4/n)
We can rewrite this as:
[(4/n) / (1+(4/n))] + [(4/n) / (1+(8/n))] + [(4/n) / (1+(12/n))] + ... + [(4/n) / (1+(4n/n))]
To simplify this further, we can notice that the denominator in each term is a telescoping series. As n approaches infinity, the terms will cancel each other except for the first and last term. Therefore, only the first and last term will contribute to the sum.
The first term [(4/n) / (1+(4/n))] simplifies to 4/(n+4), and the last term [(4/n) / (1+(4n/n))] simplifies to 4/(n+1). Therefore, the entire expression simplifies to:
4/(n+4) + 4/(n+1)
Since this expression represents the right Riemann sum for the definite integral, we can equate it to the integral to determine the upper limit of integration (b).
∫[1, b] f(x) dx = 4/(n+4) + 4/(n+1)
Now, we need to match the integrand in the expression with the function f(x).
4/(n+4) + 4/(n+1) is equal to ∫[1, b] 4/5 dx
Therefore, the upper limit of integration (b) is the same as the upper limit of the partition, which is n. The integrand is f(x) = 4/5.
To determine the upper limit of integration, we need to analyze the expression in the right Riemann sum.
The given expression in the form of a right Riemann sum is:
(1/1 + (4/n))(4/n) + (1/1 + (8/n))(4/n) + (1/1 + (12/n))(4/n) + ... + (1/1 + (4n/n))(4/n)
Let's simplify the expression step by step:
= [(1 + (4/n))/(1)]*(4/n) + [(1 + (8/n))/(1)]*(4/n) + [(1 + (12/n))/(1)]*(4/n) + ... + [(1 + (4n/n))/(1)]*(4/n)
= (4/n)*[(1 + 4/n) + (1 + 8/n) + (1 + 12/n) + ... + (1 + 4n/n)]
= (4/n)*(1 + 4/n + 1 + 8/n + 1 + 12/n + ... + 1 + 4n/n)
= (4/n)*(n/4 + n/2 + 3n/4 + ... + n)
= (4/n)*(n(1/4 + 1/2 + 3/4 + ... + 1))
= (4/n)*(n*(n/4)*(1 + n/2)/2)
= n*(n/4)*(1 + n/2)/2
Now, let's compare this expression with the general form of the right Riemann sum for a definite integral:
Riemann Sum = Σ(f(ci)*Δx)
We can see that the expression matches the form Σ(f(ci)*Δx), where c represents a value within each subinterval.
By comparing the integrand of the given expression, f(x), with the expression n*(n/4)*(1 + n/2)/2, we can determine that:
f(x) = x*(1/4 + 1/2 + 3/4 + ... + 1)
Considering the definite integral in the form ∫[1, b] f(x) dx, the upper limit of integration should be b = n.
Therefore, the upper limit of integration is b = n and the integrand is f(x) = x*(1/4 + 1/2 + 3/4 + ... + 1).