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Calculus

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Binomial Series

  • Real numbers: \(x\), \(n\)
    Whole numbers: \(m\), \(n\)
    Number of \(m\)-combinations of \(n\) elements: \(\left( {\begin{array}{*{20}{c}} n\\ m \end{array}} \right)\)
    1. The binomial series is the Maclaurin series expansion of the function \({\left( {1 + x} \right)^n}\) and, in general, is written as
      \({\left( {1 + x} \right)^n} =\) \(\sum\limits_{m = 0}^n {\left( {\begin{array}{*{20}{c}} n\\ m \end{array}} \right){x^m}} =\) \( 1 + \left( {\begin{array}{*{20}{c}} n\\ 1 \end{array}} \right)x \) \(+\; \left( {\begin{array}{*{20}{c}} n\\ 2 \end{array}} \right){x^2} + \left( {\begin{array}{*{20}{c}} n\\ 3 \end{array}} \right){x^3} + \ldots\) \(+\;\left( {\begin{array}{*{20}{c}} n\\ m \end{array}} \right){x^m} + \ldots ,\)
      where \(\left( {\begin{array}{*{20}{c}} n\\ m \end{array}} \right)\) are the binomial coefficients, \(m\) is a whole number, \(x\) is a real (or complex) variable, \(n\) is a real (or complex) power.
    2. The binomial coefficients are expressed by the formula
      \(\left( {\begin{array}{*{20}{c}} n\\ m \end{array}} \right) =\) \({\large\frac{{n\left( {n – 1} \right) \ldots \left( {n – m + 1} \right)}}{{m!}}\normalsize}\), where \(0 \le m \le n\).
    3. The binomial series converges under the following conditions (assuming that \(x\) and \(n\) are real numbers):
      \(-1 \lt x \lt 1\), if \(n \lt -1\);
      \(-1 \lt x \le 1\), if \(-1 \lt n \lt 0\);
      \(-1 \le x \le 1\), if \(n \gt 0\).
    4. Binomial theorem
      In the case of nonnegative integer powers \(n\), the binomial series is a finite sum of \({n + 1}\) terms and is referred to as binomial theorem:
      \({\left( {1 + x} \right)^n} =\) \(\sum\limits_{m = 0}^n {\left( {\begin{array}{*{20}{c}} n\\ m \end{array}} \right){x^m}} =\) \( 1 + \left( {\begin{array}{*{20}{c}} n\\ 1 \end{array}} \right)x + \left( {\begin{array}{*{20}{c}} n\\ 2 \end{array}} \right){x^2} \) \(+\;\left( {\begin{array}{*{20}{c}} n\\ 3 \end{array}} \right){x^3} + \ldots\) \(+\; \left( {\begin{array}{*{20}{c}} n\\ m \end{array}} \right){x^m} + \ldots \) \(= {1 + nx + {\large\frac{{n\left( {n – 1} \right)}}{{2!}}\normalsize} {x^2} + \ldots }\) \(+\;{ {x^n}}\)
    5. Binomial coefficients as the number of combinations The coefficients in the binomial theorem are equal to the number of \(m\)-combinations of \(n\) elements:
      \(\left( {\begin{array}{*{20}{c}} n\\ m \end{array}} \right) =\) \({\large\frac{{n!}}{{m!\left( {n – m} \right)!}}\normalsize} =\) \({\large\frac{{n\left( {n – 1} \right)\left( {n – 1} \right) \ldots \left( {n – m + 1} \right)}}{{m!}}\normalsize}.\)
    6. Some common binomial expansions:
    7. \({\large\frac{1}{{1 + x}}\normalsize} =\) \(1 – x + {x^2}\) \(-\; {x^3} + \ldots,\) \(\left| x \right| \lt 1\)
    8. \({\large\frac{1}{{1 – x}}\normalsize} =\) \(1 + x + {x^2}\) \(+\; {x^3} + \ldots,\) \(\left| x \right| \lt 1\)
    9. \(\sqrt {1 + x} =\) \(1 + {\large\frac{x}{2}\normalsize} – {\large\frac{{{x^2}}}{{2 \cdot 4}}\normalsize}\) \(+\;{\large\frac{{1 \cdot 3{x^3}}}{{2 \cdot 4 \cdot 6}}\normalsize} \) \(-\;{\large\frac{{1 \cdot 3 \cdot 5{x^4}}}{{2 \cdot 4 \cdot 6 \cdot 8}}\normalsize} + \ldots,\) \(\left| x \right| \le 1\)
    10. \(\sqrt[\large 3\normalsize]{{1 + x}} =\) \(1 + {\large\frac{x}{3}\normalsize} – {\large\frac{{1 \cdot 2{x^2}}}{{3 \cdot 6}}\normalsize} \) \(+\;{\large\frac{{1 \cdot 2 \cdot 5{x^3}}}{{3 \cdot 6 \cdot 9}}\normalsize}\) \(-\;{\large\frac{{1 \cdot 2 \cdot 5 \cdot 8{x^4}}}{{3 \cdot 6 \cdot 9 \cdot 12}}\normalsize} + \ldots,\) \(\left| x \right| \le 1\)