Properties of Infinite Series

Number sequences: $$\left\{ {{a_n}} \right\},$$ $$\left\{ {{b_n}} \right\}$$
First terms of series: $${a_1}$$, $${b_1}$$
$$N$$th terms of series: $${a_n}$$, $${b_n}$$
$$N$$th partial sum of a series: $${S_n}$$
Number of terms of a series: $$n$$
Infinite series: $$L$$, $$A$$, $$B$$
Real number: $$c$$
Continuous function: $$f\left( x \right)$$
Independent variable: $$x$$
1. Definition of an infinite series
Let $$\left\{ {{a_n}} \right\}$$ be a number sequence. An infinite series is the infinite sum of the form
$$\sum\limits_{n = 1}^\infty {{a_n}} = {a_1} + {a_2} + \ldots$$ $$+\; {a_n} + \ldots$$
2. $$N$$th partial sum of a series
$${S_n} = \sum\limits_{i = 1}^n {{a_i}} =$$ $${a_1} + {a_2} + \ldots + {a_n}$$
3. Convergence of an infinite series
An infinite series converges to $$L$$ if its partial sums $${S_n}$$ converge to $$L$$ as $$n \to \infty$$:
$$\sum\limits_{n = 1}^\infty {{a_n}} = L$$, if $$\lim\limits_{n \to \infty } {S_n} = L.$$
4. $$N$$th term test
If the series $$\sum\limits_{n = 1}^\infty {{a_n}}$$ converges, then $$\lim\limits_{n \to \infty } {a_n} = 0$$. The converse of the statement is false.
5. Sufficient condition for divergence
If $$\lim\limits_{n \to \infty } {a_n} \ne 0$$ or this limit does not exist, then the series $$\sum\limits_{n = 1}^\infty {{a_n}}$$ diverges.
6. Linear properties of convergent series
Let $$\sum\limits_{n = 1}^\infty {{a_n}} = A$$ and $$\sum\limits_{n = 1}^\infty {{b_n}} = B$$ be convergent series. Then the following properties are valid:
$$\sum\limits_{n = 1}^\infty {\left( {{a_n} + {b_n}} \right)} = A + B,\;$$ $$\sum\limits_{n = 1}^\infty {c{a_n}} = cA,$$
where $$c$$ is a real number.
7. Comparison tests
Let $$\sum\limits_{n = 1}^\infty {{a_n}}$$ and $$\sum\limits_{n = 1}^\infty {{b_n}}$$ be two infinite series such that $$0 \lt {a_n} \le {b_n}$$ for all $$n.$$ Then the following comparison tests are valid:
– If $$\sum\limits_{n = 1}^\infty {{b_n}}$$ is convergent, then $$\sum\limits_{n = 1}^\infty {{a_n}}$$ is also convergent;
– If $$\sum\limits_{n = 1}^\infty {{a_n}}$$ is divergent, then $$\sum\limits_{n = 1}^\infty {{b_n}}$$ is also divergent.
8. Limit comparison tests
Let $$\sum\limits_{n = 1}^\infty {{a_n}}$$ and $$\sum\limits_{n = 1}^\infty {{b_n}}$$ be two infinite series such that $${a_n}$$ and $${b_n}$$ are positive for all $$n.$$ Then the following limit comparison tests are valid:
– If $$0 \lt \lim\limits_{n \to \infty } {\large\frac{{{a_n}}}{{{b_n}}}\normalsize} \lt \infty ,$$ then $$\sum\limits_{n = 1}^\infty {{a_n}}$$ and $$\sum\limits_{n = 1}^\infty {{b_n}}$$ are both convergent or both divergent;
– If $$\lim\limits_{n \to \infty } {\large\frac{{{a_n}}}{{{b_n}}}\normalsize} = 0$$, then $$\sum\limits_{n = 1}^\infty {{b_n}}$$ convergent implies that series $$\sum\limits_{n = 1}^\infty {{a_n}}$$ is also convergent;
– If $$\lim\limits_{n \to \infty } {\large\frac{{{a_n}}}{{{b_n}}}\normalsize} = \infty$$, then $$\sum\limits_{n = 1}^\infty {{b_n}}$$ divergent implies that the series $$\sum\limits_{n = 1}^\infty {{a_n}}$$ is also divergent.
9. $$P$$-series
The $$p$$-series (or hyperharmonic series) $$\sum\limits_{n = 1}^\infty {{\large\frac{1}{{{n^p}}}}\normalsize}$$ converges for $$p \gt 1$$ and diverges for $$0 \lt p \le 1.$$
10. Integral test
Let $$f\left( x \right)$$ be a function which is continuous, positive and decreasing for all $$x \ge 1$$. Then the series
$$\sum\limits_{n = 1}^\infty {f\left( n \right)} = f\left( 1 \right) + f\left( 2 \right)$$ $$+\; f\left( 3 \right) + \ldots$$ $$+\; f\left( n \right) + \ldots$$
converges if the the improper integral $${\large\int\limits_1^\infty\normalsize} {f\left( x \right)dx}$$ converges, and diverges if $${\large\int\limits_1^\infty\normalsize} {f\left( x \right)dx} \to \infty .$$
11. Ratio test
Let $$\sum\limits_{n = 1}^\infty {{a_n}}$$ be a series with positive terms. Then the following rules are valid:
– If$$\lim\limits_{n \to \infty } {\large\frac{{{a_{n + 1}}}}{{{a_n}}}\normalsize} \lt 1$$, then the series $$\sum\limits_{n = 1}^\infty {{a_n}}$$ is convergent;
– If $$\lim\limits_{n \to \infty } {\large\frac{{{a_{n + 1}}}}{{{a_n}}}\normalsize} \gt 1$$, then the series $$\sum\limits_{n = 1}^\infty {{a_n}}$$ is divergent;
– If $$\lim\limits_{n \to \infty } {\large\frac{{{a_{n + 1}}}}{{{a_n}}}\normalsize} = 1$$, then the series $$\sum\limits_{n = 1}^\infty {{a_n}}$$ may converge or diverge and the ratio test is inconclusive; some other tests must be used.
12. Root test
Let $$\sum\limits_{n = 1}^\infty {{a_n}}$$ be a series with positive terms. According to the root test:
– If $$\lim\limits_{n \to \infty } \sqrt[n]{{{a_n}}} \lt 1$$, then the series $$\sum\limits_{n = 1}^\infty {{a_n}}$$ is convergent;
– If $$\lim\limits_{n \to \infty } \sqrt[n]{{{a_n}}} \gt 1$$, then the series $$\sum\limits_{n = 1}^\infty {{a_n}}$$ is divergent;
– If $$\lim\limits_{n \to \infty } \sqrt[n]{{{a_n}}} = 1$$, then the series may converge or diverge, but no conclusion can be drawn from this test.