Infinite Series
Let (a_n) be a sequence of real numbers. A partial sum of (a_n) is given by
a_1 + a_2 + \cdots a_k
for some k \leq n.
A partial sum
Let (a_n) be a sequence of real numbers. Let the sequence (s_n)
If (s_n) converges to a real number s, we say the series \sum^\infty_{n=1}a_n is convergent and write
\sum^\infty_{n=1}a_n = s.
If (s_n) does not converge, the series \sum^\infty_{n=1}a_n is divergent. If
\begin{align*} \lim_{n \to \infty} s_n &= +\infty \quad \text{then} \quad \sum^\infty_{n=1}a_n = +\infty \quad \text{ and}\\ \lim_{n \to \infty} s_n &= -\infty \quad \text{then} \quad \sum^\infty_{n=1}a_n = -\infty. \end{align*}
The harmonic series is defined
\sum^\infty_{n=1} \frac{1}{n}
and has the partial sums s_n = 1 + 1/2 + \cdots + 1/n. The sequence (s_n) is divergent, and thus the harmonic series is divergent.
The series
\sum^\infty_{n=1} \frac{1}{n(n+1)}
has the partial sum
s_n = \frac{1}{2} + \frac{1}{6} + \frac{1}{12} + \cdots + \frac{1}{n(n+1)}
Let \sum a_n = s and \sum b_n = t. Then
(a) \sum (a_n + b_n) = s + t and
(b) \sum (ka_n) = ks for every k \in \mathbb{R}.
If \sum a_n is a convergent series, then \lim_{n \to \infty} a_n = 0.
If \sum a_n is a convergent series, then the sequence of partial sums (s_n) must have a finite limit. Since a_n = s_n - s_{n-1}, we have \lim a_n = \lim s_n - \lim s_{n-1} = 0.
The infinite series \sum a_n converges if and only if for each \varepsilon > 0, there exists a natural number N such that if n \leq m \leq N, then
|a_m + a_{m+1} + \cdots + a+{n}| = |s_n - s_{m-1}| < \varepsilon.
Later.
Let \sum a_n and \sum b_n be infinite series such that a_n,b_n \leq 0 for all n. Then
(a) If \sum a_n converges and 0 \leq b_n \leq a_n for all n, then \sum b_n converges.
(b) If \sum a_n = +and 0 \leq a_n \leq b_n for all n, then \sum b_n = + \infty.
(a) Because s_{a_n} is monotone and bounded, s_{b_n} converges.
If \sum |a_n| converges, then \sum a_n is absolutly convergent, or that the sum converges absolutly.
If a series converges absolutly, then it converges.
If \sum a_n, a_n \not = 0
(a) If \limsup |a_{n+1}/a_n| < 1, the the series is absolutly convergent.
(b) If \liminf |a_{n+1}/a_n| > 1 then the series diverges.
Let f be a continuous function defined on [0,\infty) and suppose that f is positive and decreasing. Then the series \sum f(n) converges if and only if
\lim_{n \to \infty} \left( \int^n_1 f(x) \,dx \right)
If (a_n) is a decreasing sequence of positive numbers and \lim a_n = 0, then the series \sum (-1)^{n}a_n converges.