4
E n^2/2
n=0
If I remember correctly, this is how you solve this problem...
0/2 + 1/2 + 4/2 + 9/2 + 16/2 = 15
3
E 1/n^3
n=1
1/1 + 1/8 + 1/27= 251/216
You have it. There are other ways also. For instance, if you had a sum from n=1 to 9054, summing might be onerous. But for small indicies, summing is ok. Normally in ASCII, we write it as
Sigma (n=1,4) n2/2
To solve the first problem, we can use the formula for the sum of squares. The formula states that the sum of the squares of the first n natural numbers can be calculated using the equation n(n+1)(2n+1)/6.
In this case, we have n=4. Plugging that into the formula, we get:
4(4+1)(2(4)+1)/6
= 4(5)(9)/6
= 180/6
= 30
So, the sum of the squares of the first 4 natural numbers is 30.
Now, let's move on to the second problem. We are given the series 1/1 + 1/8 + 1/27 and we need to find its sum.
To find the sum, we can use the formula for the sum of a geometric series. The formula states that the sum of a geometric series can be calculated using the equation a(1 - r^n)/(1 - r), where a is the first term, r is the common ratio, and n is the number of terms.
In this case, our first term is 1/1, r is 1/2 (since each subsequent term is halved), and we have 3 terms.
Plugging these values into the formula, we get:
(1/1)(1 - (1/2)^3)/(1 - 1/2)
= (1)(1 - 1/8)/(1/2)
= (1)(7/8)/(1/2)
= 7/8 * 2/1
= 14/8
= 7/4
So, the sum of the series 1/1 + 1/8 + 1/27 is 7/4.