Formulas and Tables

Calculus

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.
  1. 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.
  1. \(P\)-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 \(0 \lt p \le 1.\)
  2. 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 \({\large\int\limits_1^\infty\normalsize} {f\left( x \right)dx} \to \infty .\)
  3. 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.
  1. 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.