Increasing and Decreasing Sequences
- A sequence \(a_n\) is increasing if \(a_{n+1} \ge a_n\) for all \(n\).
- A sequence \(a_n\) is strictly increasing if \(a_{n+1} > a_n\) for all \(n\).
- A sequence \(a_n\) is decreasing if \(a_{n+1} \le a_n\) for all \(n\).
- A sequence \(a_n\) is strictly decreasing if \(a_{n+1} < a_n\) for all \(n\).
- A sequence \(a_n\) is eventually increasing, or increasing for all \(n \ge n_0\), if \(a_{n+1} \ge a_n\) for all \(n \ge n_0\).
- A sequence \(a_n\) is eventually decreasing, or decreasing for all \(n \ge n_0\), if \(a_{n+1} \le a_n\) for all \(n \ge n_0\).
Monotone Sequences
- A sequence is called monotone if it is either increasing or decreasing.
- A sequence is called strictly monotone if it is either strictly increasing or strictly decreasing.
- A sequence is called eventually monotone if it is either eventually increasing or eventually decreasing.
Examples
- \(a_n = \frac{n}{n+1}\)
- \(b_n = \frac{1}{n}\)
- \(c_0 = 1\) and \(c_n = c_{n-1} + \frac{1}{n}\) for \(n > 0\)
- \(d_n = (-1)^n\)
- \(e_n = \cos n\)
- \(f_n = n^2\)
- \(d_n = (-2)^n\)
Bounded sequences
- A sequence \(a_n\) is bounded from above by \(B\) if \(a_n \le B\) for each \(n\).
- A sequence \(a_n\) is bounded from below by \(b\) if \(a_n \ge b\) for each \(n\).
- A sequence is bounded if it is bounded from below and bounded from above.
Examples
- \(a_n = \frac{n}{n+1}\)
- \(b_n = \frac{1}{n}\)
- \(c_0 = 1\) and \(c_n = c_{n-1} + \frac{1}{n}\) for \(n > 0\)
- \(d_n = (-1)^n\)
- \(e_n = \cos n\)
- \(f_n = n^2\)
- \(d_n = (-2)^n\)
An Example
\(a_0 = 3\) and \(\displaystyle a_n = 3 - \frac{2}{a_{n-1}}\) for \(n > 0\).
Limits of Sequences
Definition: We way that an infinite sequence \(a_n\) has a limit \(L\), or converges to \(L\), if for every \(\varepsilon > 0\) there exists an integer \(N\) such that \(\left\lvert a_n - L\right\rvert < \varepsilon\) for each \(n \ge N\).
Notation: \(\displaystyle \lim_{n\to\infty} a_n = L\)
If a sequence has a limit, we say that it is convergent.
Otherwise, we say that it is divergent.
Calculating Limits
- Limit laws
- If \(a_n = f(n)\) and \(\displaystyle \lim_{x\to\infty} f(x) = L\) then \(\displaystyle \lim_{n\to\infty} a_n = L\).
- Note that the converse is not true!
- If \(b_n = f\left(a_n\right)\) and \(\displaystyle \lim{n\to\infty} a_n = A\) and \(f\) is continuous at \(A\), then \(\displaystyle\lim_{n\to\infty} b_n =
f(A)\).
- Squeeze Theorem
Examples
- \(\displaystyle a_n = \frac{1}{n}\)
- \(\displaystyle b_n = \frac{n+1}{n}\)
- \(\displaystyle c_n = \frac{n^2}{e^n}\)
- \(\displaystyle d_n = \cos\left(\frac{1}{n}\right)\)
- \(\displaystyle e_n = \frac{1}{n!}\)
- \(\displaystyle f_n = \frac{n!}{n^n}\)
Estimates
If \(a_n\) and \(b_n\) are convergent sequences, then:
If \(a_n \le b_n\) for each \(n\), then \(\displaystyle\lim_{n\to\infty} a_n \le
\lim_{n\to\infty} b_n\).
If \(a_n < b_n\) for each \(n\), then \(\displaystyle\lim_{n\to\infty} a_n \le
\lim_{n\to\infty} b_n\).
It is still true if the inequality only holds for every \(n \ge n_0\) for some integer \(n_0\).
If a convergent sequence \(a_n\) is bounded with a lower bound \(b\) and an upper bound \(B\), then \(\displaystyle b \le \lim_{n\to\infty} a_n \le B\).
A Theorem
- Every convergent sequence is bounded.
- A bounded sequence does not have to be convergent!
- Every bounded and monotonic or eventually monotonic sequence is convergent.
An Example
\(a_0 = 3\) and \(\displaystyle a_n = 3 - \frac{2}{a_{n-1}}\) for \(n > 0\).
Infinite Series
Definition: Given an infinite sequence \(a = (a_n)_{n=k,\dots,\infty}\), we define the infinite series associated with \(a\) the formal sum \[a_k + a_{k+1} + a_{k+2} + \cdots = \sum_{i = k}^\infty a_i\] In other words, we place a \(+\) sign between each two consecutive terms of the sequence.
Question: Can this be interpreted as an actual sum, in other words, can we actually add all infinitely many terms of the sequence?
Example
\(\displaystyle a_n = \frac{1}{2^n}\) for \(n = 1, 2, dots\).
Sequence of Partial Sums
Given an infinite series \(\displaystyle\sum_{i = k}^\infty a_i = a_k + a_{k+1} + a_{k+2} + \cdots\), we define a sequence of partial sums of the series, \(s_n\) for \(n = k, k+1,
\dots\) : \[s_n = \sum_{i = k}^n a_i = a_k + a_{k+1} + a_{k+2} + \cdots + a_n\]
If the sequence \(s_n\) converges, we say that the series \(\displaystyle \sum_{i = k}^\infty a_i\) is convergent, and we define \[\displaystyle \sum_{i = k}^\infty a_i = \lim_{n\to\infty} s_n\]
Otherwise, we say that the series diverges or is divergent.
Example
\(\displaystyle \sum_{i=1}^\infty\frac{1}{2^i}\)