Cheat_Sheet

Taylor’s series:Expansions:f(x) =f(a)+(x − a)f ′ (a)+Theoretical Computer Science Cheat Sheet(x − a)2f ′′ (a)+···=2Series∞∑ (x − a) if (i) (a).i!i=011 − x=1+x + x 2 + x 3 + x 4 + ··· =11 − cx=1+cx + c 2 x 2 + c 3 x 3 + ··· =11 − x n =1+x n + x 2n + x 3n + ··· =x(1 − x) 2 = x +2x 2 +3x 3 +4x 4 + ··· =x k dndx n ( 11 − x)= x +2 n x 2 +3 n x 3 +4 n x 4 + ··· =∞∑x i ,i=0∞∑c i x i ,i=0∞∑x ni ,i=0∞∑ix i ,i=0∞∑i n x i ,∞∑e x =1+x + 1 2 x2 + 1 x i6 x3 + ··· =i! ,i=0∞∑ln(1 + x) = x − 1 2 x2 + 1 3 x3 − 1 4 x4 i+1 xi−··· = (−1)i ,i=11∞∑ln= x + 11 − x2 x2 + 1 3 x3 + 1 x i4 x4 + ··· =i ,i=1∞∑sin x = x − 1 3! x3 + 1 5! x5 − 1 7! x7 + ··· = (−1) i x 2i+1(2i + 1)! ,cos x =1− 1 2! x2 + 1 4! x4 − 1 6! x6 + ··· =tan −1 x = x − 1 3 x3 + 1 5 x5 − 1 7 x7 + ··· =(1 + x) n =1+nx + n(n−1)2x 2 + ··· =1(1 − x) n+1 =1+(n +1)x + ( )n+22 x 2 + ··· =xe x − 1=1− 1 2 x + 112 x2 − 1720 x4 + ··· =12x (1 − √ 1 − 4x) = 1+x +2x 2 +5x 3 + ··· =121√ 1 − 4x=1+x +2x 2 +6x 3 + ··· =( √1 1 − 1 − 4x√ 1 − 4x 2x11 − x ln 11 − x(ln11 − x) n=1+(2+n)x + ( )4+n2 x 2 + ··· == x + 3 2 x2 + 11 6 x3 + 2512 x4 + ··· =) 2= 1 2 x2 + 3 4 x3 + 1124 x4 + ··· =x1 − x − x 2 = x + x 2 +2x 3 +3x 4 + ··· =F n x1 − (F n−1 + F n+1 )x − (−1) n x 2 = F n x + F 2n x 2 + F 3n x 3 + ··· =i=0i=0∞∑i=0(−1) i x2i(2i)! ,∞∑(−1) i x 2i+1(2i +1) ,∞∑( ) nx i ,ii=0i=0∞∑( ) i + nx i ,i∞∑ B i x i,i!∞∑( )1 2ix i ,i +1 i∞∑( ) 2ix i ,i∞∑( ) 2i + nx i ,i∞∑H i x i ,i=0i=0i=0i=0i=0i=1∞∑ H i−1 x i,i∞∑F i x i ,i=2i=0∞∑F ni x i .i=0Ordinary power series:∞∑A(x) = a i x i .i=0Exponential power series:∞∑ x iA(x) = a ii! .i=0Dirichlet power series:∞∑ a iA(x) =i x .i=1Binomial theorem:n∑( n(x + y) n = xk)n−k y k .k=0Difference of like powers:n−1∑x n − y n =(x − y) x n−1−k y k .k=0For ordinary power series:∞∑αA(x)+βB(x) = (αa i + βb i )x i ,x k A(x) =i=0∞∑a i−k x i ,i=kA(x) − ∑ k−1i=0 a ix ix k =A(cx) =A ′ (x) =∞∑a i+k x i ,i=0∞∑c i a i x i ,i=0∞∑(i +1)a i+1 x i ,i=0∞∑xA ′ (x) = ia i x i ,i=1∫∞∑ a i−1A(x) dx = x i ,ii=1A(x)+A(−x)∞∑= a 2i x 2i ,2i=0A(x) − A(−x)∞∑= a 2i+1 x 2i+1 .2i=0Summation: If b i = ∑ ij=0 a i thenB(x) = 11 − x A(x).Convolution:⎛∞∑ i∑A(x)B(x) = ⎝i=0j=0a j b i−j⎞⎠ x i .God made the natural numbers;all the rest is the work of man.– Leopold Kronecker