Series Formulas - mathportal.org

Series Formulas 1. Arithmetic and Geometric Series Definitions: First term: a1 Nth term: an Number of terms in the Series : n Sum of the first n terms: Sn Difference between successive terms: d Common ratio: q Sum to infinity: S Arithmetic Series Formulas : ()11naand=+ 112iiiaaa ++= 12nnaaSn+= ()1212nandSn+ = Geometric Series Formulas : 11nnaa q = 11iiiaaa += 11nna q aSq = ()111nna qSq = 1111foaSqrq <<= 2. Special Power Series Powers of Natural Numbers ()1112nkkn n==+ () ()2111 2 16nkkn nn==++ ()2321114nkkn n==+ Special Power Series ()2311..11:1x xxxxfor= + +++ < < ()2311..11:1x xxxxfor= + + < <+ 231..2! 3!xxxex= + +++ ()()2345ln 1..112 3 4 5:xxxxxorxxf+= + + < < 3579sin..3! 5! 7! 9!xxxxx x= + + 2468cos 1..2! 4! 6! 8!xxxxx= + + 3572 .3 15 31522:forxxxx xx = ++++ < < 3579sinh..3! 5! 7! 9!xxxxx x= ++++ 2468cosh 1.

R Lagrange s form a x n f x x a R Cauch s form a x n ξ ξ ξ ξ ξ − − = ≤ ≤ − − = ≤ ≤ − This result holds if f(x) has continuous derivatives of order n at last. If lim 0n n R →∞ =, the infinite series obtained is called Taylor series for f(x) about x = a. If a = 0 the series is often called a Maclaurin series. Binomial ...

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