Formulas and Tables

Calculus

Power Series Expansions

Function: \(f\left( x \right)\)
Real numbers: \(x\), \(a\), \(\xi\)

Remainder term: \({R_n}\)
Whole numbers: \(n\)

  1. Taylor series
    If a function \(f\left( x \right)\) has continuous derivatives up to \(\left( {n + 1} \right)\)th order inclusive, then this function can be expanded in a power series about the point \(x = a\) by the Taylor formula:
    \(f\left( x \right) =\) \(\sum\limits_{n = 0}^\infty {{f^{\left( n \right)}}\left( a \right){\large\frac{{{{\left( {x – a} \right)}^n}}}{{n!}}\normalsize}} =\) \({f\left( a \right) + f’\left( a \right)\left( {x – a} \right)} \) \(+\;{{\large\frac{{f^{\prime\prime}\left( a \right){{\left( {x – a} \right)}^2}}}{{2!}}\normalsize} + \ldots} \) \(+\;{{\large\frac{{{f^{\left( n \right)}}\left( a \right){{\left( {x – a} \right)}^n}}}{{n!}}\normalsize} + {R_n}},\)
    where the remainder term \({R_n}\) in the Lagrange term is given by the expression
    \({R_n} = {\large\frac{{{f^{\left( {n + 1} \right)}}\left( \xi \right){{\left( {x – a} \right)}^{n + 1}}}}{{\left( {n + 1} \right)!}}\normalsize},\) \(a \lt \xi \lt x.\)
    If this expansion converges over a certain range of \(x\) centered at \(a\), that is, \(\lim\limits_{n \to \infty } {R_n} = 0\), then the expansion is called Taylor series of the function \(f\left( x \right)\) expanded about the point \(a\).
  2. A Maclaurin series is a special case of a Taylor series when the power series expansion is performed at the point \(a = 0:\)
    \(f\left( x \right) = \sum\limits_{n = 0}^\infty {{f^{\left( n \right)}}\left( 0 \right){\large\frac{{{x^n}}}{{n!}}}\normalsize} =\) \(f\left( 0 \right) + f’\left( 0 \right)x \) \(+\;{\large\frac{{f”\left( 0 \right){x^2}}}{{2!}}\normalsize} + \ldots\) \(+\;{\large\frac{{{f^{\left( n \right)}}\left( 0 \right){x^n}}}{{n!}}\normalsize} + {R_n}.\)

    Below are some important Maclaurin series expansions.

  3. \({e^x} = 1 + x + {\large\frac{{{x^2}}}{{2!}}\normalsize} \) \(+\;{\large\frac{{{x^3}}}{{3!}}\normalsize} + \ldots\) \(+\;{\large\frac{{{x^n}}}{{n!}}\normalsize} + \ldots \)
  4. \({a^x} = 1 + {\large\frac{{x\ln a}}{{1!}}\normalsize} \) \(+\;{\large\frac{{{{\left( {x\ln a} \right)}^2}}}{{2!}}\normalsize} + {\large\frac{{{{\left( {x\ln a} \right)}^3}}}{{3!}}\normalsize} + \ldots\) \(+\;{\large\frac{{{{\left( {x\ln a} \right)}^n}}}{{n!}}\normalsize} + \ldots \)
  5. \(\ln \left( {1 + x} \right) =\) \(x – {\large\frac{{{x^2}}}{2}\normalsize} + {\large\frac{{{x^3}}}{3}\normalsize} \) \(-\;{\large\frac{{{x^4}}}{4}\normalsize} + \ldots\) \(+\;{\large\frac{{{{\left( { – 1} \right)}^n}{x^{n + 1}}}}{{n + 1}}\normalsize} \pm \ldots ,\) \( – 1 \lt x \le 1\)
  6. \(\ln {\large\frac{{1 + x}}{{1 – x}}\normalsize} =\) \(2\Big( {x + {\large\frac{{{x^3}}}{3}\normalsize} + {\large\frac{{{x^5}}}{5}\normalsize} }\) \({+\;{\large\frac{{{x^7}}}{7}\normalsize} + \ldots } \Big),\) \(\left| x \right| \lt 1\)
  7. \(\ln x =\) \(2\Big[ {{\large\frac{{x – 1}}{{x + 1}}\normalsize} + {\large\frac{1}{3}\normalsize} {{\left( {\large\frac{{x – 1}}{{x + 1}}\normalsize} \right)}^3} }\) \(+\;{ {\large\frac{1}{5}\normalsize}{{\left( {\large\frac{{x – 1}}{{x + 1}}\normalsize} \right)}^5} + \ldots } \Big],\) \(x \gt 0\)
  8. \(\cos x =\) \(1 – {\large\frac{{{x^2}}}{{2!}}\normalsize} + {\large\frac{{{x^4}}}{{4!}}\normalsize} – {\large\frac{{{x^6}}}{{6!}}\normalsize} + \ldots\) \(+\; {\large\frac{{{{\left( { – 1} \right)}^n}{x^{2n}}}}{{\left( {2n} \right)!}}\normalsize} \pm \ldots \)
  9. \(\sin x =\) \(x – {\large\frac{{{x^3}}}{{3!}}\normalsize} + {\large\frac{{{x^5}}}{{5!}}\normalsize} – {\large\frac{{{x^7}}}{{7!}}\normalsize} + \ldots\) \(+\; {\large\frac{{{{\left( { – 1} \right)}^n}{x^{2n + 1}}}}{{\left( {2n + 1} \right)!}}\normalsize} \pm \ldots \)
  10. \(\tan x =\) \(x + {\large\frac{{{x^3}}}{3}\normalsize} + {\large\frac{{2{x^5}}}{{15}}\normalsize}\) \(+\;{\large\frac{{17{x^7}}}{{315}}\normalsize} + {\large\frac{{62{x^9}}}{{2835}}\normalsize} + \ldots,\) \(\left| x \right| \lt {\large\frac{\pi }{2}\normalsize}\)
  11. \(\cot x =\) \({\large\frac{1}{x}\normalsize} – \Big( {{\large\frac{x}{3}\normalsize} + {\large\frac{{{x^3}}}{{45}}\normalsize} }\) \(+\;{ {\large\frac{{2{x^5}}}{{945}}\normalsize} + {\large\frac{{2{x^7}}}{{4725}}\normalsize} + \ldots } \Big),\) \(\left| x \right| \lt \pi \)
  12. \(\arcsin x =\) \(x + {\large\frac{{{x^3}}}{{2 \cdot 3}}\normalsize} + {\large\frac{{1 \cdot 3{x^5}}}{{2 \cdot 4 \cdot 5}}\normalsize} + \ldots\) \(+\;{\large\frac{{1 \cdot 3 \cdot 5 \ldots \left( {2n – 1} \right){x^{2n + 1}}}}{{2 \cdot 4 \cdot 6 \ldots \left( {2n} \right)\left( {2n + 1} \right)}}\normalsize} + \ldots,\) \(\left| x \right| \lt 1\)
  13. \(\arccos x =\) \({\large\frac{\pi }{2}\normalsize} – \Big( {x + {\large\frac{{{x^3}}}{{2 \cdot 3}}\normalsize} + {\large\frac{{1 \cdot 3{x^5}}}{{2 \cdot 4 \cdot 5}}\normalsize} + \ldots}\) \({+\;{\large\frac{{1 \cdot 3 \cdot 5 \ldots \left( {2n – 1} \right){x^{2n + 1}}}}{{2 \cdot 4 \cdot 6 \ldots \left( {2n} \right)\left( {2n + 1} \right)}}\normalsize} + \ldots} \Big),\) \(\left| x \right| \lt 1\)
  14. \(\arctan x =\) \(x – {\large\frac{{{x^3}}}{3}\normalsize} + {\large\frac{{{x^5}}}{5}\normalsize} – {\large\frac{{{x^7}}}{7}\normalsize} + \ldots\) \(+\;{\large\frac{{{{\left( { – 1} \right)}^n}{x^{2n + 1}}}}{{2n + 1}}\normalsize} \pm \ldots ,\) \(\left| x \right| \le 1\)
  15. \(\cosh x =\) \(1 + {\large\frac{{{x^2}}}{{2!}}\normalsize} + {\large\frac{{{x^4}}}{{4!}}\normalsize} + {\large\frac{{{x^6}}}{{6!}}\normalsize} + \ldots\) \(+\;{\large\frac{{{x^{2n}}}}{{\left( {2n} \right)!}}\normalsize} + \ldots \)
  16. \(\sinh x =\) \(x + {\large\frac{{{x^3}}}{{3!}}\normalsize} + {\large\frac{{{x^5}}}{{5!}}\normalsize} + {\large\frac{{{x^7}}}{{7!}}\normalsize} + \ldots\) \(+\;{\large\frac{{{x^{2n + 1}}}}{{\left( {2n + 1} \right)!}}\normalsize} + \ldots \)