Binomial Expansion for Class 12 Mathematics Second part

Add. Mathematics

Grade

12

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

$\log(1+x)$

$\log(1-x)$

Lesson Binomial Expansion

By: Ghanashyam Adhikari

0. Home & Intro

1. Formula Derivation

2. Formula Bank

3. 3D & Graph Sim

4. Worked Examples (1-17)

5. Quizzes (1-10)

6. Exercise 2.1 & 2.2

7. Exam & Auto-Marking

Grade 12 Mathematics: Binomial Expansion

Welcome to the interactive learning simulation on Exponential & Logarithmic Infinite Series and Binomial Theorem Applications. Developed specifically for Nepal CDC Grade 12 Higher Secondary Mathematics Curriculum.

Infinite Series

Understand the rigorous expansion of $e^x$, $e^{-x}$, $\log(1+x)$, and $\log(1-x)$ with limit derivations.

Interactive Convergence

Visualize how Taylor polynomial truncations converge to transcendental functions in real-time.

CDC Model Exams

Practice complete exercise problems, self-assessment quizzes, and fully automated exam grading.

1. Application of Binomial Theorem & Derivations

Definition: Application of Binomial Theorem

The binomial theorem has many important applications, the most useful of which is the determination of approximate values of certain algebraically as well as arithmetical quantities and sums of certain infinite series.

As an application of the binomial theorem, we can arrive at:

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

which terminates after $n+1$ terms when $n$ is any positive integer. But when $n$ is any real number different from a positive integer, the expansion does not terminate and it is valid, only if $|x| < 1$, and this expansion is known as binomial series.

$$(1-x)^{-n} = 1 + nx + \frac{n(n+1)}{2!}x^2 + \frac{n(n+1)(n+2)}{3!}x^3 + \dots \quad (|x| < 1)$$

Exponential and Logarithmic Series

Any function of the form $y = f(x) = a^x, a > 0$ is called an exponential function in which the base $a$ is constant and the index $x$ is a variable. The inverse of an exponential function is called a logarithmic function which is denoted by $\log_a x$. So, if $y = a^x$, we have $x = \log_a y$.

There is a special type of exponential function $e^x$, where $e$ is the limiting value: $$e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$$ The value of $e$ lies between $2$ and $3$ and is approximately $2.718282$. The corresponding logarithmic function is called the natural logarithmic function and is denoted by $\log x$, base $e$ being understood. The number $e$ is known as Euler's number.

Expansion of $e^x$ (Rigorous Proof)

We prove that for all values of $x$:

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

Proof: If $n > 1$ (so that $\frac{1}{n} < 1$), then we have by binomial theorem that:

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

Hence, when $n$ is infinitely large ($n \to \infty$):

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

Putting $x = 1$, we have:

$$\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \dots = e$$

Now, $\left(1 + \frac{1}{n}\right)^{nx} = \left\{\left(1 + \frac{1}{n}\right)^n\right\}^x \to e^x$ as $n \to \infty$. Making $n$ infinitely large, we get the infinite expansion of $e^x$.

Logarithmic Series Expansion $\log(1+x)$

For $-1 < x \le 1$, the logarithmic expansion is:

$$\log(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \dots \text{ to } \infty$$

Replacing $x$ with $-x$, we get the expansion for $\log(1-x)$ for $-1 \le x < 1$:

$$\log(1-x) = -x - \frac{x^2}{2} - \frac{x^3}{3} - \frac{x^4}{4} - \dots \text{ to } \infty$$

Subtracting the two logarithmic expansions gives:

$$\log(1+x) - \log(1-x) = \log\left(\frac{1+x}{1-x}\right) = 2 \left[ x + \frac{x^3}{3} + \frac{x^5}{5} + \dots \right] \quad (|x| < 1)$$

2. Complete Master Formula Bank

All standard formulas required for Grade 12 Binomial, Exponential, and Logarithmic Series problems:

Combination Identity

$${}^nC_r = \frac{n!}{r!(n-r)!}$$

Combinations of $n$ items taken $r$ at a time.

Symmetry Property

$${}^nC_r = {}^nC_{n-r}$$

Complementary combinations are equal.

General Term in $(a+b)^n$

$$T_{r+1} = {}^nC_r a^{n-r} b^r$$

Finds $(r+1)^{\text{th}}$ term in positive integer expansion.

Exponential Constant $e$

$$e = 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \dots \approx 2.71828$$

Euler's number infinite series limit.

Even Terms of $e$

$$\frac{e + e^{-1}}{2} = 1 + \frac{1}{2!} + \frac{1}{4!} + \dots$$

Hyperbolic cosine series $\cosh(1)$.

Odd Terms of $e$

$$\frac{e - e^{-1}}{2} = \frac{1}{1!} + \frac{1}{3!} + \frac{1}{5!} + \dots$$

Hyperbolic sine series $\sinh(1)$.

Logarithmic Series $\log(1+x)$

$$\log(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \dots \quad (-1 < x \le 1)$$

Natural log expansion with alternating signs.

Logarithmic Series $\log(1-x)$

$$\log(1-x) = -x - \frac{x^2}{2} - \frac{x^3}{3} - \dots \quad (-1 \le x < 1)$$

All terms negative for negative argument.

Logarithmic Difference

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

Rapidly converging series for computing logs.

2D Taylor Polynomial Convergence

Chart.js Engine

Select function and number of terms ($N$) to see truncation approximation vs exact curve:

Target Function: Exponential e^x Exponential e^(-x) Logarithmic ln(1+x) Logarithmic ln(1-x)

Number of Terms ($N$): 4

Observed Error: Calculating...

3D Infinite Expansion Surface

Three.js WebGL

Interactive 3D plot of surface $z = f(x, y) = e^x \cos(y)$ expanded via bivariate series. Drag mouse to rotate view!

Drag to rotate | Scroll to zoom

Rotation Controls:

4. Worked Examples (1 to 17)

Complete step-by-step solutions for Grade 12 examination problems. Click any problem to view solution details:

5. Concept Quizzes (1 to 10)

Test your conceptual understanding of binomial, exponential, and logarithmic series.

Score: 0 / 10

6. CDC Textbook Exercises

Exercise 2.1 Focus:

Exponential Series $e^x$, $e^{-x}$, and sums of factorial reciprocal series.

Exercise 2.2 Focus:

Logarithmic Series $\log(1+x)$, $\log(1-x)$, and general binomial approximations.

7. CDC Grade 12 Model Mock Exam

Timed practice test with automated instant grading and diagnostic performance analytics.

15:00

Grade 12 Higher Secondary Mathematics Assessment

This exam contains 10 standardized questions covering Exponential Series, Logarithmic Expansion, and Binomial Theorem applications. Complete within the time limit and click "Submit & Grade Exam".

Examination Diagnostic Report

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