Question

What does it mean for a sequence to converge?

Answer 1

In this problem, we have to find whether it means for a sequence to converge. We should know that a sequence that converges is one that adds to a number and a sequence is said to be convergent if it approaches some limit or if it ends up to a particular number to which it tends.

Complete step by step answer:
We have to find whether it means for a sequence to converge.
We know that, formally a sequence converges to the limit, we can write it as,
$\Rightarrow \displaystyle \lim_{n \to \infty }{{S}_{n}}=S$
Where, if any $\varepsilon >0$, there exist an $N$ such that $\left| {{S}_{n}}-S \right|<\varepsilon$ for $n > N$. If ${{S}_{n}}$ does not converge then it is said to diverge.
We can say that, where every bounded monotonic sequence converges and every unbounded sequence diverges.
We should know that a sequence converges when it keeps getting closer and closer to a certain value.
We can now take an example sequence.
Where, the terms of $\dfrac{1}{n}$ are: $1,\dfrac{1}{2},\dfrac{1}{3},\dfrac{1}{4}$ and so on.
We can see that the sequence converges to 0 as it gets closer and closer to 0 and it is also said to be a convergent sequence.
Therefore, a sequence converges if it approaches some limit or if it ends up to a particular number to which it tends.

Note: We should always remember that a sequence that converges is one that adds to a number and a sequence is said to be convergent if it approaches some limit or if ends in up to a particular number to which it tends and every bounded monotonic sequence converges and every unbounded sequence diverges.