Mathematics

Taylor and Maclaurin Series

Definition of Taylor series: f(x)=f(a)+f′(a)(x−a)+f′′(a)(x−a)22!+⋯+f(n−1)(a)(x−a)n−1(n−1)!+Rn Rn=f(n)(ξ)(x−a)nn! where a≤ξ≤x, ( Lagrangue’s form ) Rn=f(n)(ξ)(x−ξ)n−1(x−a)(n−1)! where a≤ξ≤x, ( Cauch’s form ) This result holds if f(x) has continuous derivatives of order n at last. If limn→+∞Rn=0, the...

Power Series

Powers of Natural Numbers ∑k=1nk=12n(n+1) ∑k=1nk2=16n(n+1)(2n+1) ∑k=1nk3=14n2(n+1)2 Special Power Series 11−x=1+x+x2+x3+⋯(for −1<x<1) 11+x=1−x+x2−x3+⋯(for −1<x<1) ex=1+x+x22!+x33!+⋯ ln(1+x)=x−x22+x33−x44+⋯(for −1<x<1) sinx=x−x33!+x55!−x77!+⋯ cosx=1−x22!+x44!−x66!+⋯ tanx=x−x33+2x515−17x7315+⋯(for −π2<x<π2) sinhx=x+x33!+x55!+x77!+⋯ coshx=1+x22!+x44!+x66!+⋯

Arithematic and Geometric Series

Notation: Number of terms in the series: n First term: a1 Nth term: an Sum of the first n terms: Sn Difference between successive terms: d Common ratio: q...

Limits Formulas

The General Limit Formulas If limx→af(x)=l and limx→ag(x)=m ,then limx→a [f(x)±g(x)]=l±m limx→a [f(x)⋅g(x)]=l⋅m limx→af(x)g(x)=lm limx→ac⋅f(x)=c⋅l limx→a1f(x)=1l The Common Limits limx→∞ (1+1n)n=e limx→∞ (1+n)1/n=e limx→0 sinxx=1 limx→0 tanxx=1 limx→0 cosx−1x=0 limx→a xn−anx−a=nan−1

Hyperbolic Formulas

Definitions of hyperbolic functions sinhx=ex−e−x2 coshx=ex+e−x2 tanhx=ex−e−xex+e−x=sinhxcoshx cschx=2ex−e−x=1sinhx sechx=2ex+e−x=1coshx cothx=ex+e−xex−e−x=coshxsinhx Derivatives ddxsinhx=coshx ddxcoshx=sinhx ddxtanhx=sech2x ddxcschx=−cschx⋅cothx ddxsechx=−sechx⋅tanhx ddxcothx=−csch2x Hyperbolic identities cosh2x−sinh2x=1 tanh2x+sech2x=1 coth2x−csch2x=1 sinh(x±y)=sinhx⋅coshy±coshx⋅sinhy cosh(x±y)=coshx⋅coshy±sinhx⋅sinhy sinh(2⋅x)=2⋅sinhx⋅coshx cosh(2⋅x)=cosh2x+sinh2x sinh2x=−1+cosh2x2 cosh2x=1+cosh2x2 Inverse...

Trigonometric Formulas

Right-Triangle Definitions sinα=OppositeHypotenuse cosα=AdjacentHypotenuse tanα=OppositeAdjacent cscα=1sinα=HypotenuseOpposite secα=1cosα=HypotenuseAdjacent cotα=1tanα=AdjacentOpposite Reduction Formulas sin(−x)=−sin(x) cos(−x)=cos(x) sin(π2−x)=cos(x) cos(π2−x)=sin(x) sin(π2+x)=cos(x) cos(π2+x)=−sin(x) sin(π−x)=sin(x) cos(π−x)=−cos(x) sin(π+x)=−sin(x) cos(π+x)=−cos(x) Basic Identities sin2x+cos2x=1 tan2x+1=1cos2x cot2x+1=1sin2x Sum and Difference Formulas...

Logarithm Formulas

Logarithm formulas y=logax⟺ay=x  (a,x>0,a≠1) loga1=0 logaa=1 loga(mn)=logam+logan logamn=logam−logan logamn=n⋅logam logam=logbm⋅logab logam=logbmlogba logab=alogba logax=lnalnx

Roots Formulas

Notation: a,b : bases (a≥0,b≥0  if  n=2k) n,m: powers Formulas (a√n)n=a (a√n)m=am−−−√n a√n−−−√m=a√nm (am−−−√n)p=anp−−−√n am−−−√n=anp−−−√np 1a√n=an−1−−−−√na ab−−√n=a√n⋅b√n ab−−√n=a√nb√n a√nb√m=ambn−−−√nm a√n⋅b√m=ambn−−−−√nm a±b√−−−−−−√=a+a2−b−−−−−√2−−−−−−−−−−√±a−a2−b−−−−−√2−−−−−−−−−−√ 1a√±b√=a√∓b√a−b

Exponential Formulas

Exponential Formulas ap=a⋅a⋅…ap   (if  p∈N) a0=1   (if   a≠0) ar⋅as=ar+s aras=ar−s (ar)s=ar⋅s (a⋅b)r=ar⋅br (ab)r=arbr a−r=1ar ars=ar−−√s

Algebra Formulas part 3

Quadric Equation: ax2+bx+c=0 Solutions (roots): x1,2=−b±b2−4ac−−−−−−−√2a If D=b2−4ac is the discriminant , then the roots are 1. real and unique if D>0 2. real and equal if D=0 3....