Sum of series?

Question

Hi.
I have the following problems that I keep doing wrong for some reason. Can someone please walk me through the solutions step by step?
1) Sum of 2j^2 where the first term is 1 and last 4. The answer is 60 but I keep getting 68. The formula I use is n/2 (a1 + an)
2) Sum of 1/k where the first term is 3 and last is 6.
Please walk me through this step by step so I can see where I went wrong.
Thank you.

Answer 1

Question 1 : You are trying to use the formula for the sum of the integers, when what you want is a formula for the sum of the squares of the integers :

Sum of the squares of the integers from 1 to n =

n/6(n + 1)(2n + 1)

Since you appear to want twice this sum ( sum of 2j^2 is the same as twice the sum of j^2, since the initial "2" is a common factor in every term, therefore it can be taken outside the summation ), then the formula becomes

2*n/6(n +1)(2n+1) = n/3(n + 1)(2n + 1)

Since your last term has n = 4, all you have to do is to substitute 4 for n :

4/3(5)(9) = 60


If you want the sum for a different range of numbers, say from n = 5 to n = 10, then first work out the sum for 1 to 10 by putting n = 10; then find the sum for 1 to 4 by putting n = 4 ; subtract the second sum from the first to get the required answer.



Question 2 : The series 1/1 + 1/2 + 1/3 + .........1/k is called the Harmonic Series and there is no simple formula for finding the sum of a given number of terms. In fact, the sum of the series is infinite, it goes on growing as you extend the series further and further. If you do a search for the Harmonic Series you will find thousands of interesting results concerning it.

If you simply want the sum of the four terms

1/3 + 1/4 + 1/5 + 1/6 , then you will just have to use normal arithmetic :

Sum = (20 + 15 + 12 + 10 ) / 60 = 57 / 60

Answer 2

2j^2: numbers are 2: 8: 18: 32 which add up to 60 - you are using the formula for the sum of an arithmetic progression and this is NOT an arithmetic progression (You are actually adding 2 + 12 + 22 + 32 to get to 68)

1/k = 1/3 + 1/4 + 1/5 + 1/6 = (20 +15 +12+10)/60 = 57/60

Answer 3

4
∑2j^2 =
j=1
4
2∑j^2 = 2(4)(4 + 1)(8 + 1)/6 =
j=1
2*4*5*9/6 = 4*5*3 = 60

1/3 + 1/4 + 1/5 + 1/6 =
(20 + 15 + 12 + 10)/60 =
57/60 = 0.95