Series Formulas - mathportal.org

1 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.

2 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..2! 4! 6! 8!xxxxx= ++++ 3572 15 31522:forxxxx xx = + + < < 3. Taylor and Maclaurin Series Definition: ()()()()112( )( )()( ) ( ) ( )..2!1 !nnnfa x af a x af xf af a x aRn =+ ++++ ( )( ) ()( )( ) ()()()1'!'1 !nnnnnnfx aRLagrange s formaxnfxx aRCauch s formaxn = = This result holds if f(x) has continuous derivatives of order n at last. If lim0nnR =, 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 .

3 Binomial Series ()()()()12 23 312 23 !3!..123nnnnnnnnnn nn nna xana xaxaxnnnaa xaxax +=++++ =++++ Special cases: () xxxx += + + << ()223411 2 35 ..141xxxxxx += + +< < ()323411 3 6 10 15 . += + +< < ()12321 1 31 3 2 42 4 611xxxxx += + + < ()1232111 2 42 4 611xxxxx += + ++ < Series for exponential and logarithmic functions ! 3!xxxex= + +++ ()()23lnln1 !3!xx ax aax a= ++++ ()234ln 3 411xxxxxx+= +< ()231 11 11ln +=+++ Series for trigonometric functions ! 5! 7!xxxxx= + + 246cos !

4 4! 6!xxxx= + + ()()222 13572 2 12 15 3152 !22nnnnB xxxxxxnx = +++++ << ()22 45 94205!nnnB xxxxxxxn = << ()2246561sec 24 72202 !2nnE xxxxxxn = ++++ <<++ ()()2232 2 36020!nnnE xxxxxnx =+++++<< 357111 31 3 3 2 4 5 214 6 71xxxxxx = + + ++ << 351111 3 21415xxxxxx = = + + + < < 5 71 xif xxxx + + << = + + ++ < .23 5 71 xxxxxxxxxxxxxi xxfx + + = <<= + + < + + + Series for hyperbolic functions ! 5! 7!xxxxx= ++++ 246cosh ! 4! 6!xxxx= ++++ ()()()12 22 13571 2 2 3!nnnnnB xxxxnxxxif = + +<+ <+ ()()122 1371 45 94520!

5 NnnniB xxfxxxxxn =+ ++<+<+