Calculate \(\displaystyle \int_{-2}^1 \frac{1}{x^2}\;dx\)
Calculate \(\displaystyle \int_{-2}^1 \frac{1}{x^{1/3}}\;dx\)
P-integrals
\[\int_0^1 \frac{1}{x^p}\;dx\]
P-integrals
\[\int_1^\infty \frac{1}{x^p}\;dx\]
Comparison Theorem
Does \(\displaystyle \int_1^\infty \frac{1}{xe^x}\;dx\) converge?
Comparison Theorem
Does \(\displaystyle \int_e^\infty \frac{\ln x}{x}\;dx\) converge?
Infinite Sequences
Sequence:
- A (possibly infinite) list of numbers
- A function from (a subset of) integers to real numbers
Examples
- \(a_n = 1 + 2^n\), \(n = 0, 1, 2, \ldots\)
- \(\left\{\frac{1}{n+1}\right\}_{n=1, 2, \ldots}\)
- \(\left\{\frac{1}{n}\right\}_{n=2, 3, \ldots}\)
- \(c_0 = 2\) and \(c_n = 2c_{n-1} + 3\) for \(n > 0\)
Arithmetic Sequences
\(a_n = a_{n-1} + d\) for \(n > 0\)
Geometric Sequences
\(g_n = r\cdot g_{n-1}\) for \(n > 0\)
Factorials
- \(a_0 = 1\)
- \(a_n = n\cdot a_{n-1}\) for \(n > 0\)
- \(d_0 = 1\)
- \(d_1 = 1\)
- \(d_n = n\cdot a_{n-2}\) for \(n > 1\)
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.
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.
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.