Series Formulas

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Arithmetic Series

\( S_n = a_1 + a_2 +...+ a_n \)
\( \quad \quad = a + (a + d) + (a + 2d) + ... + a + (n-1)d \)
\( d \) is the common difference
\( a_1 = a \) , first term in the series
\( a_n = a + (n - 1)d \) , nth term in the series
\( S_n = \dfrac{n}{2} (\text{first term} + \text{last term}) \)
\( \quad \quad = \dfrac{n}{2}(a_1 + a_n) \)
\( \quad \quad = \dfrac{n}{2}[ 2 a + (n-1)d ] \)

Geometric Series

\( S_n = a_1 + a_2 +...+ a_n \)
\( \quad \quad = a + a \cdot r + a \cdot r^2 + ... + a \cdot r^{n-1} \)
\( r \) is the common ratio
\( a_1 = a \) , first term in the series
\( a_n = a \cdot r^{n-1} \) , nth term in the series
\( S_n = a_1\dfrac{1 - r^n}{1 - r} \)
\( \quad \quad = a \dfrac{1 - r^n}{1 - r} \)
For an infinite gemetric series and for \( |r| \lt 1 \)
\( S_{\infty} = \dfrac{a_1}{1 - r} = \dfrac{a}{1 - r} \)

Integer Series

\( \sum_{k=1}^{n} k = 1 + 2 + 3...+n = \dfrac{1}{2} n(n+1)\)
\( \sum_{k=1}^{n} k^2 = 1^2 + 2^2 + 3^2...+n^2 = \dfrac{1}{6} n(n+1)(2n+1)\)
\( \sum_{k=1}^{n} k^3 = 1^3 + 2^3 + 3^3...+n^3 = [ \dfrac{1}{2} n(n+1) ] ^2\)

Binomial Theorem (Binomial Expansion)

\( \displaystyle (x+y)^n = \sum_{k=0}^{n} {n \choose k} x^{n-k} y^k\)
\( \displaystyle \quad \quad = {n \choose 0} x^{n} + {n \choose 1} x^{n-1} y^1 + {n \choose 2} x^{n-2} y^2 + ... + {n \choose n} y^{n} \)
where the coefficient \( {n \choose k} \) represents the number of (distinct) combinations of \( k \) object that can be formed from a sample of \( n \) objects and is given by
\( \displaystyle {n \choose k} = \dfrac{n!}{k!(n-k)!} \)

Taylor Series

A Taylor series expansion of a function for values close to \( x = a \) is given by
\( f(x) = f(a) + (x-a) f'(a) + \dfrac{(x-a)^2}{2!} f"(a) + \dfrac{(x-a)^3}{3!} f^{(3)}(a) + ... + \dfrac{(x-a)^n}{n!} f^{(n)}(a)+...\)

Maclaurin Series

A Taylor series expansion of a function \( f \) for values close to \( x = 0 \) is called Maclaurin series of function \( f \) and is obtained be setting \( a = 0 \) in Taylor series which gives
\( f(x) = f(0) + x f'(0) + \dfrac{x^2}{2!} f"(0) + \dfrac{x^3}{3!} f^{(3)}(0) + ... + \dfrac{x^n}{n!} f^{(n)}(0)+...\)

Maclaurin Series of Some Common Functions

\( e^x = 1 + x + \dfrac{x^2}{2!} + ... + \dfrac{x^n}{n!} + ... \) all \( x \)
\( \ln(1+x) = x - \dfrac{x^2}{2} + \dfrac{x^3}{3} - ... + (-1)^{n+1}\dfrac{x^n}{n} + ... \) \(| x| \le 1 ; x \ne -1 \)
\( \sin x = \dfrac{e^{jx} - e^{-jx}}{2j} = x - \dfrac{x^3}{3!} + \dfrac{x^5}{5!} - \dfrac{x^7}{7!}... \) all \( x \)
\( \cos x = \dfrac{e^{jx} + e^{-jx}}{2} = 1 - \dfrac{x^2}{2!} + \dfrac{x^4}{4!} - \dfrac{x^6}{6!}... \) all \( x \)
\( \tan x = x + \dfrac{1}{3} x^3 + \dfrac{2}{15} x^5 + \dfrac{17}{315}x^7... \) \( |x| \lt \pi/2 \)
\( \sinh x = \dfrac{e^{x} - e^{-x}}{2} = x + \dfrac{x^3}{3!} + \dfrac{x^5}{5!} + \dfrac{x^7}{7!}... \) all \( x \)
\( \cosh x = \dfrac{e^{x} + e^{-x}}{2} = 1 + \dfrac{x^2}{2!} + \dfrac{x^4}{4!} + \dfrac{x^6}{6!}... \) all \( x \)
\( \tanh x = x - \dfrac{1}{3} x^3 + \dfrac{2}{15} x^5 - \dfrac{17}{315}x^7... \) \( |x| \lt \pi/2 \)
\( \sin^{-1} x = \arcsin x = x + \dfrac{1 \cdot x^3}{2\cdot3} + \dfrac{1 \cdot 3 x^5}{2 \cdot 4 \cdot 5} + \dfrac{1 \cdot 3 \cdot 5 x^7}{2 \cdot 4 \cdot 6 \cdot 7} ... \) \( |x| \lt 1 \)
\( \tan^{-1} x = \arctan x = x - \dfrac{x^3}{3} + \dfrac{x^5}{5} +... \) \( |x| \lt 1 \)
\( \sinh^{-1} x = \text{arcsinh} \; x = x - \dfrac{1 x^3}{2\cdot3} + \dfrac{1 \cdot 3 x^5}{2 \cdot 4 \cdot 5} - \dfrac{1 \cdot 3 \cdot 5 x^7}{2 \cdot 4 \cdot 6 \cdot 7} ... \) \( |x| \lt 1 \)
\( \tanh^{-1} x = \arctan x = x + \dfrac{x^3}{3} + \dfrac{x^5}{5} +... \) \( |x| \lt 1 \)