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Taylor and Maclaurin Series

Definition of Taylor series: \( f(x) = f(a) + f'(a)(x-a) + \frac{f”(a)(x-a)^2}{2!} + \cdots +\frac{f^{(n-1)}(a)(x-a)^{n-1}}{(n-1)!} + R_n \) \( R_n = \frac{f^{(n)}(\xi)(x-a)^n}{n!} \text{ where } a \leq \xi \leq x, \quad \text{ ( Lagrangue’s form )} \) \( R_n = \frac{f^{(n)}(\xi)(x-\xi)^{n-1}(x-a)}{(n-1)!} \text{ where } a \leq \xi \leq x, \quad \text{ ( Cauch’s form )} \) This result […]

Special Power Series

Powers of Natural Numbers \( \sum\limits_{k=1}^n k = \frac{1}{2}n(n+1) \) \( \sum\limits_{k=1}^n k^2 = \frac{1}{6}n(n+1)(2n+1) \) \( \sum\limits_{k=1}^n k^3 = \frac{1}{4}n^2(n+1)^2 \) Special Power Series \( \frac{1}{1-x} = 1 + x + x^2 +x^3 + \cdots \quad(\text{for } -1 < x < 1) \) \( \frac{1}{1+x} = 1 – x + x^2 – x^3 + \cdots \quad(\text{for } -1 […]

Arithmetic and Geometric Series

Notation: Number of terms in the series: n First term: \(a_1\) \(N^{th}\) term: \(a_n\) Sum of the first n terms: \(S_n\) Difference between successive terms: d Common ratio: q Sum to infinity: S Arithmetic Series Formulas: \( a_n = a_1 + (n-1)d \) \( a_i = \frac{a_{i-1} + a_{i+1}}{2} \) \( S_n = \frac{a_1 + a_n}{2} \cdot […]