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 […]

Definite Integrals of Logarithmic Functions

  \( \int^1_0 x^m(\ln x)^n dx = \frac{(-1)^n n!}{(m+1)^{n+1}} , \quad m>-1,\, n=0,1,2,\dots \)   \( \int^1_0 \frac{\ln x}{1+x} dx = -\frac{\pi^2}{12} \)   \( \int^1_0 \frac{\ln x}{1-x}dx = -\frac{\pi^2}{6} \)   \( \int^1_0 \frac{\ln (1+x)}{x} dx = \frac{\pi^2}{12} \)   \( \int^1_0 \frac{\ln (1 – x)}{x} dx = – \frac{\pi^2}{6} \)   \( \int^1_0 \ln x \ln (1+x) \,dx […]

How to Teach Calculus

When we talk about calculus, we must know that calculus deals with derivatives, infinite series, integrals and limits. All this, people learn in the first year of college. There is debate how calculus should be taught in high school because calculus have many advance over anything that has been taught before, but there is a […]

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