Evaluate
lim (1³ +2³ +3³ +…+ n3)/n^4
n →∞
by showing that the limit is a particular definite integral and evaluating that definite integral.
sum(1) = 1^3/1^4 = 1 or 4/4
sum(2) = (1 + 8)/16 = 9/16
sum(3) = (1+8+27)/81 = 36/81
= 4/9 or 16/36
sum(4) = (1+8+27+64)/256
= (1+8+27+64)/256
= 100/256 = 25/64
sum(5) = (1+8+27+64+125)/625
= 225/625 = 9/25 = 36/100
I noticed I can change my numerator to (n+1)^2 for sum(n)
and the denominators are
1,16,36,64,100
or 4(1^2), 4(2^2), 4(3^2), 4(5^2)
ahhh, so we can say:
(1³ +2³ +3³ +…+ n3)/n^4 = (n+1)^2/(4n^2)
you might want to check this for one or more sums
so lim(1³ +2³ +3³ +…+ n3)/n^4 as x --->∞
= lim (n+1)^2 / (4n^2) as x ---> ∞
= lim (n^2 + 2n + 1)/4n^2 as x ---> ∞
= lim ( 1/4 + 1/(2n) + 1/(4n^2) as x ---> ∞
= 1/4
To evaluate the given expression
lim (1³ + 2³ + 3³ + … + n³) / n^4
n →∞
we can rewrite it in terms of a sum and then convert it into a definite integral.
Step 1: Rewrite the expression using the summation notation:
lim ∑(k³) / n^4
n →∞ k=1
Step 2: Simplify the expression by factoring out the common term:
lim ∑ k³ / n³ * n / n³
n →∞ k=1
Step 3: Rewrite the expression in terms of a Riemann sum by dividing each term by n^4:
lim ∑ (k/n)³ * (1/n)
n →∞ k=1
Step 4: Recognize the Riemann sum as a particular definite integral:
lim ∑ (k/n)³ * (1/n)
n →∞ k=1
This sum represents the Riemann sum of the function f(x) = x³ over the interval [0, 1].
Step 5: Express the sum as a definite integral:
integral from 0 to 1 of x³ dx
Step 6: Evaluate the definite integral:
To calculate the definite integral ∫ x³ dx, we can use the power rule:
integral of x^n dx = (x^(n+1)) / (n+1)
Applying the power rule, we have:
integral of x³ dx = (x^(3+1)) / (3+1)
= (x^4) / 4
Now, we evaluate the integral from 0 to 1:
[ (1^4) / 4 ] - [ (0^4) / 4 ]
= (1/4) - (0/4)
= 1/4
Therefore, the evaluated definite integral is 1/4.
To evaluate the given limit, we will first rewrite the expression as a definite integral and then find the value of that integral.
Let's start by expressing the sum of cubes as a definite integral. We can begin by noting that the sum of cubes from 1 to n can be represented as:
1³ + 2³ + 3³ + ... + n³ = ∫₀ⁿ x³ dx.
Now, let's rewrite the given expression using this representation:
lim (1³ + 2³ + 3³ + ... + n³) / n^4
n → ∞
= lim (∫₀ⁿ x³ dx) / n^4
n → ∞
Now we need to evaluate this definite integral. Let's calculate it:
∫₀ⁿ x³ dx = [1/4 x^4] from 0 to n
= (1/4 n^4) - (1/4 * 0^4)
= (1/4 n^4)
Now, substituting this result back into the expression, we have:
lim (∫₀ⁿ x³ dx) / n^4
n → ∞
= lim (1/4 n^4) / n^4
n → ∞
= lim 1/4
n → ∞
= 1/4
Therefore, the value of the given limit is 1/4.