Hi! Looking to find
n∑k=1 8(k/n)
(n is supposed to be on top, k=1 below the sigma sign)in closed form.
I've tried doing 8*[(n(n+1)/2)/n], and that didn't seem to work. I'm not sure if I'm supposed to do anything with the n or if it might be an issue with what to do with the division of k/n. Thanks!
I got it!
n is a constant, so it's just 8/n * n(n+1)/2.
To find the closed form expression for the given series,
n∑k=1 8(k/n),
let's break it down step by step.
First, let's simplify the expression inside the parentheses, which is (k/n).
To do this, we can distribute the 8 to both k and n:
8(k/n) = (8/n)k
Next, we can bring the constant coefficient (8/n) out of the sigma (∑) notation:
n∑k=1 (8/n)k
Using the properties of sigma notation, we can rewrite the above expression as:
(8/n) ∑k=1 nk
Now, let's focus on the sigma notation:
∑k=1 nk
This is a series that sums up the terms nk for k = 1 to n.
We can convert the above series into a closed form expression using the formula for the sum of an arithmetic series:
∑k=1 nk = (n/2)(1 + n)
Now, let's substitute this back into our expression:
(8/n) ∑k=1 nk = (8/n) * (n/2)(1 + n)
Now, let's simplify this expression further:
= 4(1 + n)
Therefore, the closed form expression for the given series is:
n∑k=1 8(k/n) = 4(1 + n)
So, the closed form expression for the series is 4(1 + n).