Question

Can someone explain why an infinite series won't change even if you add another term of the sequence to the end of it?

Answer 1

As much mathematical evidence as possible, if any, please.

Answer 2

what about the infinite series that you meant?

Answer 3

ex. IF: S=(1-1+1-1+1-1+1.....)
S=(1-1+1-1+1-1+1.....)+1

Answer 4

well, infinite series has no end, so you can't add any term at the end.

Answer 5

(1-1+1-1+1-1+1.....)+1 would not be infinite

Answer 6

What if I make
S=(1-1+1-1+1-1+1.....)
1-S=1-(1-1+1-1+1-1+1.....) ----> 1-S=1-1+1-1+1....

Does 1-S=S?

Answer 7

if you can explain how one adds a term to the end of an INFINITE series

Answer 8

there is no *end* so this question is meaningless

Answer 9

In that case, is that the reason that Grandi's Series is wrong?

Answer 10

no, Grandi's series diverges because the partial sums never settle. when we write \(S=1+-1+1+-1+\dots\) what we mean is \(S=\lim_{n\to\infty}\underbrace{(1+-1+1+-1+\dots)}_{n\text{ terms}}\)

Answer 11

don't forget that the "sum" of an infinite series is the limit of the partial sums
that is what it means
since the partial sums in this case are \(1,0,1,0,1,...\) there is no limit
hence no "sum" because they mean the same thing

Answer 12

yes, that's what I was saying above... and we define there to be no limit because no matter how far we go we can't make partial sums less than \(1\) apart

Answer 13

\(\displaystyle\lim_{n\to\infty}a_n=L\) means that we can pick any (i.e. arbitrarily small) \(\epsilon\) and find some \(N\) such that \(n>N\) guarantees \(|a_n-L|<\epsilon\)