Useful Identities

UI.ID.1: exponential ID

 

P.S: 

UI.ID.2: exponential ID

UI.ID.3: exponential ID

Series convolution

SC.ID.1: linear series convolution 

 

where

 

 SC.ID.2: exponential series convolution  

where

 

therefore

Taylor and Maclaurin Series

• TS.ID.1: Taylor Series Definition

Sigma representation

• TS.ID.2: Maclaurin Series
  

• TS.ID.3: Trigonometric Functions

proof

---

proof

---

proof

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• TS.ID.4: Hyberbolic Functions

proof

---

proof

---

proof

TS.ID.5:  Geometric Series

TS.ID.6:  Logarithemtic Function

Exponential Function

• E.ID.1: Taylor Series

• E.ID.2: Euler's Formula

• E.ID.3: Bernouli's Formula
  
  
  

  

  
  

   Proof
  
  power  expansion
  
  

  

  
  

   

  after dividing  

  

  
  

  again , after dividing  

  
  

   

  

  
  

  

  
  

  
  
• E.ID.4: Logarithms
  

• E.ID.5: Hyberbolic Functions
  

Geometric Series

• GS.ID.1: Geometric Series Formula
  

Proof

Multiply s with r , therefore

Subtract rs from s , therefore

therefore,

As n goes to infinity, the absolute value of r must be less than one for the series to converge, therefore ,

• GS.ID.2: Geometric Series of Exponential

As n goes to infinity, the absolute value of e converges

• GS.ID.3: Geometric Series Derivatives

Given

(1)

First derivative

(2)

Second derivative

(3)

(4)

(5)

from (3) and (5)

(6)

from (2) and (6) 

 

(7)

 

 (8)