Binomial Expansion and Logarithmic Series for Class 12 Mathematics 2nd part

Mathematics

Grade 12

$e^x$ and $e^{-x}$

$\log(1+x)$

$\log(1-x)$

Lesson

Binomial, Exponential & Logarithmic Series

By: Ghanashyam Adhikari

NEB Grade 12 Curriculum Lab

Binomial, Exponential & Logarithmic Series

Welcome to your digital mathematics laboratory designed specifically for Grade 12 students. Every textbook worked example (1 to 17) and Exercise 2.2 problem (1 to 11 with all subparts) is fully solved with extremely detailed, step-by-step teacher explanations tailored for basic learners. Explore rigorous formula derivations ($e^x$, $\log(1+x)$), test convergence on the live visualizer with $e^{-x}$ and $\log(1-x)$, practice interactive quizzes with immediate feedback, and take the official CDC Nepal Model Exam (50 Marks)!

Exponential Series

Master $e^x$, $e^{-x}$, and Euler's constant $e \approx 2.71828$ with step derivations.

Logarithmic Series

Explore $\log(1+x)$, $\log(1-x)$, and symmetric expansions with deep clarity.

Exercise 2.2 Bank

Problems 1 to 11 fully solved with elementary classroom explanations.

Curriculum Derivations

Step-by-Step Formula Derivations for Grade 12

Explained thoroughly just like your teacher writes on the classroom whiteboard for basic-level learners.

Derivation 1: Expansion of $e^x$ (Exponential Series)

Teacher's Explanation (Basic Level): We know from algebra that any exponential function is written as $y = a^x$ where $a > 0$. When the base is Euler's number $e$, defined as $\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$, we write $e^x$. Let us use the binomial theorem for positive integer $n$:

$$\left(1 + \frac{1}{n}\right)^{nx} = 1 + nx \cdot \frac{1}{n} + \frac{nx(nx-1)}{2!} \cdot \frac{1}{n^2} + \frac{nx(nx-1)(nx-2)}{3!} \cdot \frac{1}{n^3} + \dots$$

Simplifying the terms by taking $n$ inside each bracket factor:

$$= 1 + x + \frac{x\left(x - \frac{1}{n}\right)}{2!} + \frac{x\left(x - \frac{1}{n}\right)\left(x - \frac{2}{n}\right)}{3!} + \dots$$

Now, taking the limit as $n \to \infty$, terms like $\frac{1}{n}, \frac{2}{n}$ approach $0$:

$$e^x = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots + \frac{x^r}{r!} + \dots$$

Derivation 2: Expansion of $\log_e(1+x)$ (Logarithmic Series)

Teacher's Explanation (Basic Level): For any number $x$ lying between $-1$ and $+1$ ($|x| < 1$), the logarithmic series gives the sum to infinity.

Starting from the general binomial expansion and integrating or applying logarithmic properties (as proven in higher calculus), we establish:

$$\log_e(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \dots + (-1)^{r-1}\frac{x^r}{r} + \dots$$

Important Note: This series is valid for $-1 < x \le 1$. If we put $x = 1$, we get $\log_e 2 = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \dots$.

Core Theory

Binomial Theorem for Any Rational Index

$$(1+x)^m = 1 + mx + \frac{m(m-1)}{2!}x^2 + \frac{m(m-1)(m-2)}{3!}x^3 + \dots + \frac{m(m-1)\dots(m-r+1)}{r!}x^r + \dots$$

Exponential Series ($e^x$ & $e^{-x}$)

$$e^x = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots$$

$$e^{-x} = 1 - \frac{x}{1!} + \frac{x^2}{2!} - \frac{x^3}{3!} + \dots$$

The Logarithmic Series ($\log(1+x)$ & $\log(1-x)$)

$$\log_e(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \dots$$

$$\log_e(1-x) = -x - \frac{x^2}{2} - \frac{x^3}{3} - \frac{x^4}{4} - \dots$$

$$\log_e\left(\frac{1+x}{1-x}\right) = 2\left(x + \frac{x^3}{3} + \frac{x^5}{5} + \dots\right)$$

Formula Reference & Interactive Lab

Select any vital formula to test parameter values and approximate series sums interactively.

Exponential

Expansion of $e^x$

Interactive Parameter Test

Value of $x$: 1.0

Terms ($N$): 5

Series Sum Approximation

2.71666

Exact Value

2.71828

Interactive Series Convergence Visualizer

Visualize how Taylor polynomial terms converge to exact transcendental curves ($e^x, e^{-x}, \log(1+x), \log(1-x)$).

Exponential: e^x Exponential: e^-x Logarithmic: log(1+x) Logarithmic: log(1-x) Binomial: (1-x)^-1

Exact Function

Series Approximation ($N$ terms)

Approximation Terms ($N$): 4

Worked Examples Bank (Examples 1 to 17)

Extremely clear, step-by-step classroom teacher explanations for every single textbook example.

Assessment

Interactive Chapter Quizzes

Question 1 of 10

Exercise 2.2 Complete Problem Bank

All textbook questions (1 to 11 with all subparts). Click "Show Solution" to reveal step-by-step teacher explanations.

CDC Model Examination Portal

Enter the examination password to unlock the official 50-mark CDC Nepal Model Examination with Equation Editor and Image Upload capabilities.

Grade 12 Additional Mathematics - Model Examination

Full Marks: 50 | Time: 1.5 Hours | CDC Nepal Grid Compliant

54:00

Instructions: Answer all 15 questions across Group A (2 marks $\times$ 5), Group B (3 marks $\times$ 5), and Group C (5 marks $\times$ 5). You can type mathematical solutions using the equation symbols toolbar or upload handwritten solution photos (JPG/PNG/WEBP).