Some infinite series have a sum: \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots = 1, they are called convergent
Others don’t: 1 - 1 + 1 - 1 + \cdots, they are called divergent
If \sum_{n=1}^{\infty} a_n converges then a_n \to 0.
Series diverges if \lim_{n\to\infty} a_n fails to exist or is different from 0.
This however does not work the other way, if a_n \to 0 the series still can be divergent.
Partial sums form non-decreasing sequence.
These series converge iff partial sums are bounded from above.
example: Harmonic series \sum_{n=1}^{\infty} \frac{1}{n^p} is convergent iff p > 1
\sum_{n=1}^{\infty} a_n converges if there is a convergent series \sum_{n=1}^{\infty} c_n with a_n \le c_n for all n > N, for some integer N.
\sum_{n=1}^{\infty} a_n diverges if there is a divergent series \sum_{n=1}^{\infty} d_n with a_n \ge d_n for all n > N, for some integer N.
Let \lim_{n \to \infty} \frac{a_n}{b_n} = L
Measure the rate of growth or decline of a series: \frac{a_{n+1}}{a_n}
Let \rho = \lim_{n \to \infty} \frac{a_{n+1}}{a_n}, then:
Let \rho = \lim_{n \to \infty} \sqrt[n]{a_n}, then:
Series in which terms alternate between positive and negative. Usually consists of (-1)^n.
\sum (-1)^n \frac{1}{n}
The series \sum (-1)^{n+1} a_n converges if all three are satisfied:
We say that \sum a_n is convergent absolutely iff \sum |a_n| is convergent
We say that \sum a_n is convergent conditionally iff \sum a_n is convergent and \sum |a_n| is divergent
If \sum |a_n| converges, then \sum a_n converges.