How do you use the summation formulas to rewrite the expression #Sigma (4i^2(i-1))/n^4# as k=1 to n without the summation notation and then use the result to find the sum for n=10, 100, 1000, and 10000?
Let
# S_n = sum_(i=1)^n (4i^2(i-1))/n^4 #
# :. S_n = 4/n^4sum_(i=1)^n (i^3-i^2) #
# :. S_n = 4/n^4{sum_(i=1)^n i^3 - sum_(i=1)^n i^2 }#
And using the standard results: We have; And this has been calculated using Excel for
What happens as [ NB As an additional task we could possibly conclude that as Now, And so, Which confirms our assumption!
By signing up, you agree to our Terms of Service and Privacy Policy
To rewrite the expression Σ (4i^2(i-1))/n^4 as k=1 to n without summation notation, we first expand and simplify the expression:
Σ (4i^2(i-1))/n^4 = 4/n^4 * Σ (i^3 - i^2) from i=1 to n
Using the summation formulas, we get:
Σ (i^3 - i^2) = (Σ i^3) - (Σ i^2)
Now, apply the summation formulas for the sum of the first n integers (Σ i) and the sum of the squares of the first n integers (Σ i^2):
Σ i = (n(n+1))/2 Σ i^2 = (n(n+1)(2n+1))/6 Σ i^3 = [(n(n+1))/2]^2
Substitute these formulas into the expression:
Σ (4i^2(i-1))/n^4 = 4/n^4 * [(n(n+1))/2]^2 - (n(n+1)(2n+1))/6
Now, plug in the values of n to find the sum for n=10, 100, 1000, and 10000:
For n=10: Substitute n=10 into the expression and calculate the result. For n=100: Substitute n=100 into the expression and calculate the result. For n=1000: Substitute n=1000 into the expression and calculate the result. For n=10000: Substitute n=10000 into the expression and calculate the result.
By signing up, you agree to our Terms of Service and Privacy Policy
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7