Question

(Introductory Real Analysis) - This isn't actually the problem I'm being asked to do, but it's part of it; I'm trying to find the formula for the sum of all even integers, and the way I'm doing it conceptually makes sense to me, but isn't working out mathematically. Why?

Answer 1

What I'm doing is taking the formulas for both the summation of all positive integers and the summation of all even positive integers and subtracting one from the other, to hopefully yield the summation formula for the sum of all odd positive integers. Why isn't this working?
\[S_{N(even)}=n(n+1)\]\[S_{N(all)}=\frac{n(n+1)}{2}\]\[S_{N(all)}-S_{N(even)}=\frac{n(n+1)}{2}-n(n+1)=\frac{-n(n+1)}{2}=-\frac{n^2}{2}-\frac{n}{2}\]

The formula for the sum of all odd digits is n^2, what am I doing wrong?

Answer 2

How do you know the sum of even numbers is \(n(n+1)\) ?

Answer 3

don't you think it must depend on whether \(n\) is odd or even ?

Answer 4

It looks like my formula is wrong then, yeah. Don't know how I missed that.

Answer 5

S_N even is infinite set ?

Answer 6

for finite set of even that start from 1

\(S_N=\{2,4,6,8,.....,2n\}=2\{1,2,3,....,n\} \)
\(\sum S_N=2(n)(n+1)/2\)

so if this is what you mean in ur formula then it's correct

Answer 7

now what you have doing is wrong that is

sum of all integers between 2n and 1
is \(\sum S_A=(2n)(2n+1)/2\)

Answer 8

hmm it depend on how the last element looks like see in the even case last number only can be even.

Answer 9

sum of odd \(S_O\)
\(\sum S_O =2n^2+n-(n^2-n)=n^2\)

Answer 10

:O i never knew sum of odds is n^2 weird !!!

Answer 11

so what you have done wrong is note sum of all integers should be bigger than sum of only evens but look at ur summation
\(S_{N(all)}=\frac{n(n+1)}{2}<n(n+1)=S_{N(even)}\)
which is a contradiction and don't make sense :D