∑n k=1 (24k(k-1)/n^7

Question

Use the summation formulas to rewrite the expression ∑n k=1 (24k(k-1)/n^7 without the summation notation.

Answer 1

n

∑     24k(k-1)/n⁷ = 24/n⁷ ∑ k(k-1) = 24/n⁷ ∑ (k² - k) = 24/n⁷ (∑ k² - ∑ k)  -- realizing the 24/n⁷ is a constant

k=1

 

Omitted the from k =1 to n in all except the first ∑.

 

Now the key is understanding the formulas for ∑ k² and ∑ k.

 

Those are n(n+1)(2n+1)/6  and n(n+1)/2 respectively.

 

Substituting we have:

 

24/n⁷ [n(n+1)(2n+1)/6 - n(n+1)/2]

 

There are various solutions for simplifying the above:

 

8(n² - 1)/n⁶ is about the simplest. 

 

To convince yourself, let n = 3 and try using the original summation version for k = 1 to 3. Then substitute 3 for n in the final formula. The results should be the same. 

 

This exercise is likely to prepare you for using summation notation to evaluate a Riemann sum.

 

See if you can get to this: if not try to search for a file titled no summation notation: https://www.wyzant.com/resources/files/540308/no_summation_notation