Q. 3TB - Calculus

Question

Let \(\left \{ a_{n} \right \}\) be a sequence. What does it mean for the sequence to converge? What does it mean for the sequence to diverge? 

Step-by-Step Solution

Answer

Suppose \(\left \{ a_{n} \right \}\) is a sequence of real numbers. We say that \(\lim_{n\rightarrow \infty }\left \{ a_{n} \right \}=L\) for some real number \(L\), or equivalently that \(\left \{ a_{n} \right \}\rightarrow L\), if the following statement is true: 

For any \(\varepsilon >0\), there exists some \(N>0\) such that if \(n>N\), then \(\left \{ a_{n} \right \}\epsilon \left ( L-\varepsilon ,L+\varepsilon  \right )\). 

If \(\left \{ a_{n} \right \}\rightarrow L\)  for some real number \(L\), then we say that the sequence \(\left \{ a_{n} \right \}\) converges to \(L\). If no such \(L\) exists, then we say that the sequence diverges. 

Step 1. Given Information

It is given that \(\left \{ a_{n} \right \}\) is a sequence.

The objective is to what does it mean for the sequence to converge or diverge.

Step 2. Know whether the sequence is converging or diverging

Suppose \(\left \{ a_{n} \right \}\) is a sequence of real numbers. We say that \(\lim_{n\rightarrow \infty }\left \{ a_{n} \right \}=L\) for some real number \(L\), or equivalently that \(\left \{ a_{n} \right \}\rightarrow L\), if the following statement is true: 

For any \(\varepsilon >0\), there exists some \(N>0\) such that if \(n>N\), then \(\left \{ a_{n} \right \}\epsilon \left ( L-\varepsilon ,L+\varepsilon  \right )\). 

If \(\left \{ a_{n} \right \}\rightarrow L\)  for some real number \(L\), then we say that the sequence \(\left \{ a_{n} \right \}\) converges to \(L\). If no such \(L\) exists, then we say that the sequence diverges.