# Closed forms of series / sums

atrus_ovis
Hello, i just came accros:

Sum(i) , from i=1 to i=n
which apparently equals n(n+1)/2

-Is there a way to derive this from the sum, or you just have to use your intuition and think through what exactly is being summed and the range of summation?
-Do you have any resources to offer, that includes all the common sums and its closed forms?

Very elementary proof: Let S=sum,

S = 1 + 2 + ... + n
S = n + (n-1) + ... + 1
2S = (n+1) + (n+1) + ... + (n+1) {n terms} = n(n+1)

atrus_ovis
Damn... that was way too trivial, i feel worthless =P

Staff Emeritus
Homework Helper
Hello, i just came accros:

Sum(i) , from i=1 to i=n
which apparently equals n(n+1)/2

-Is there a way to derive this from the sum, or you just have to use your intuition and think through what exactly is being summed and the range of summation?
-Do you have any resources to offer, that includes all the common sums and its closed forms?

Check out the very interesting paper: www.math.uic.edu/~kauffman/DCalc.pdf
This gives some tools to derive closed-form formula's of some nice sums. And these tools are based on the normal tools of calculus.

Dr. Seafood
I think these kinds of formulas are conjectured (guessed) using a clever trick or by noticing a pattern, and then formally established using mathematical induction. In the particular example of the sum of the first n integers, take n = 10:

We want the sum 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10. Notice that 11 = 1 + 10 = 2 + 9 = 3 + 8, etc. and so we've added 11 5 = 10/2 times. The conjecture is that the sum is 11 + 11 + 11 + 11 + 11 = 5(11) = (10/2)(11). Perhaps this holds for all n: one of n or n + 1 is even since they are adjacent integers, so n(n + 1) is always divisible by two; so n(n + 1)/2 is an integer. Makes sense that the sum of the first n integers is ... We guess that 1 + 2 + ... + n = n(n + 1)/2 and establish this by induction.

This kind of guess-work is useful for a simple, intuitive arithmetic formula like this one, but probably not for more complicated expressions/conjectures.

Check out the very interesting paper: www.math.uic.edu/~kauffman/DCalc.pdf
This gives some tools to derive closed-form formula's of some nice sums. And these tools are based on the normal tools of calculus.

That is so awesome. Even as a PM major taking courses like RA, I find discrete math so beautiful -- so graph theory and combinatorics really interests me (but the combinatorics program at my school is too computer-y for me D:). It's so cool to see the methods of calculus used in a discrete setting.

Last edited:
Mensanator
Hello, i just came accros:

Sum(i) , from i=1 to i=n
which apparently equals n(n+1)/2

-Is there a way to derive this from the sum, or you just have to use your intuition and think through what exactly is being summed and the range of summation?
-Do you have any resources to offer, that includes all the common sums and its closed forms?