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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 […] ## Definite Integrals of Exponential Functions \( \int^\infty_0 e^{-ax} \cos bx \, dx = \frac{a}{a^2 + b^2}$$   $$\int^\infty_0 e^{-ax} \sin bx \, dx = \frac{b}{a^2 + b^2}$$   $$\int^\infty_0 \frac{e^{-ax} \sin bx}{x} \, dx = \arctan \frac{b}{a}$$   $$\int^\infty_0 \frac{e^{-ax}-e^{-bx}}{x} dx = \ln \frac{b}{a}$$   $$\int^\infty_0 e^{-ax^2} \, dx = \frac{1}{2} \sqrt{ \frac{\pi}{a} }$$   $$[…] ## Definite Integrals of Trigonometric Functions Basic formulas \( \int^{\pi/2}_0 \sin^2x\,dx = \int^{\pi/2}_0 \cos^2x\,dx = \frac{\pi}{4}$$   $$\int^\infty_0 \frac{\sin(px)}{x} \,dx = \left\{ \begin{array}{l l l} \pi/2 & p > 0 \\ ~0 & p = 0 \\ -\pi/2 & p < 0 \\ \end{array} \right.$$   $$\int^\infty_0 \frac{\sin^2px}{x^2} = \frac{\pi\,p}{2}$$   $$\int^\infty_0 \frac{1 – \cos(px)}{x^2}dx = \frac{\pi\,p}{2}$$ […]

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