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: \(n\)

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: \(x\)

Real number: \(c\)

Continuous function: \(f\left( x \right)\)

Independent variable: \(x\)

- Definition of an infinite series

Let \(\left\{ {{a_n}} \right\}\) be a number sequence. An infinite series is the infinite sum of the form

\(\sum\limits_{n = 1}^\infty {{a_n}} = {a_1} + {a_2} + \ldots\) \(+\; {a_n} + \ldots \) - \(N\)th partial sum of a series

\({S_n} = \sum\limits_{i = 1}^n {{a_i}} =\) \({a_1} + {a_2} + \ldots + {a_n}\) - 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 \(\lim\limits_{n \to \infty } {S_n} = L.\) - \(N\)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. - 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. - 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. - 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. - 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. - \(P\)-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 \(0 \lt p \le 1.\) - 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 \({\large\int\limits_1^\infty\normalsize} {f\left( x \right)dx} \to \infty .\) - 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. - 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.