Examples
Example 1:
\sum_{n=1}^{\infty} 1
\lim_{n \to \infty} 1 = 1
Since the limit is not zero, the series diverges.
Example 2: \sum_{n=1}^{\infty} \frac{n+1}{n}
Let's find the limit of the terms:
\lim_{n \to \infty} \frac{n+1}{n} = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right) = 1 + 0 = 1
Since the limit of the terms is
1 \neq 0
, by the nth Term Test for Divergence, the series diverges.
Example 3: \sum_{n=1}^{\infty} (-1)^n
Let's find the limit of the terms:
\lim_{n \to \infty} (-1)^n
This limit does not exist because the terms oscillate between
-1
and
1
. Thus, it is not equal to zero.
The series diverges.
Example 4: \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}
\lim_{n \to \infty} \frac{1}{\sqrt{n}} = 0
The limit of the sequence is zero, so the series may converge or diverge.
Example 5: \sum_{n=1}^{\infty} \frac{1}{n}
\lim_{n \to \infty} \frac{1}{n} = 0
The limit of the sequence is zero, so the series may converge or diverge.