- TS.ID.1: Taylor Series Definition

Sigma representation

- TS.ID.2: Maclaurin Series

- TS.ID.3: Trigonometric Functions

**proof**

**proof**

**proof**

- TS.ID.4: Hyberbolic Functions

**proof**

**proof**

**proof**

TS.ID.5:

**Geometric Series**

TS.ID.6:

**Logarithemtic Function**- TS.ID.1: Taylor Series Definition

Sigma representation

- TS.ID.2: Maclaurin Series

- TS.ID.3: Trigonometric Functions

- TS.ID.4: Hyberbolic Functions

TS.ID.5:** **Geometric Series

TS.ID.6:** **Logarithemtic 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

- 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

(1)

First derivative

(2)

Second derivative

(3)

(4)

(5)

from (3) and (5)

(6)

from (2) and (6)

(7)

(8)

1. Introduction

The generating function of Bernoulli polynomial is defined as:

2. Identities

Proof

therefore,

(1)

Proof

therefore,

(1)

multiplying and dividing by m , therefore

(2)

(3)

(4)

from (1) and (4),

(5)

therefore,

proof

therefore;

based on (UI.ID.1)

and based on

therefore,

(1)

(2)

(3)

from (1) and (3) , therefore

equating the similar coefficients

(1)

(2)

from (1) and (2)

(4)

from (3) and (4)

(5)

therefore,

(6)

(7)

(8)

(9)

subtracting (6) from (7) or (8) from (9) (above)

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