We define infinite series as summation of terms in sequence.
where sequence is defined to function from natural number to real numbers $f:\mathbb{N} \rightarrow \mathbb{R}$. But $\infty$ does not belong to natural numbers right? so what does $\infty$ as upper limit represent in this $\sum_{n=1}^{\infty} a_n$
I am confused because consider Cauchy Criterion which states as follows
A series $ \sum_{n=1}^{\infty} a_n $ converges iff $\forall \epsilon>0$ $\,$ $\exists N \in \mathbb{N} $ $\,$ $\forall n>m\geq N $ $\,$ $|\sum_{k=m+1}^{n} a_k| < \epsilon $
can I take upper limit to $n = \infty$ that is can i say $|\sum_{k=m+1}^{\infty} a_k| < \epsilon$ ?
I think answer is no because $\infty$ does not belong to natural numbers.But we seem to be doing that when we write infinite series notation
Best Answer
You are correct in your statements regarding the infinity not being a number.
So, what is it ?
We define a sequence approaching infinity if the terms grow without bound.
Same goes with series, when we add up terms of an infinite sequence we are trying to find the limit of the sequence of partial sums as the index grows without bound.
Therefore infinity is not a number but we need it to express the growth without bound.