How to Use Taylor Series to Solve Maths Problems?
The concept of Taylor Series plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Whether you’re tackling problems for CBSE, JEE, or exploring higher studies, understanding Taylor series helps you approximate and analyze complex functions efficiently.
What Is Taylor Series?
A Taylor Series is a way to represent a function as an infinite sum of terms calculated from the values of its derivatives at a single point. This powerful technique allows you to approximate functions like ex, sinx, cosx, and ln(1+x) using polynomials. You’ll find this concept applied in areas such as power series analysis, polynomial approximations, and error estimation in numerical methods.
Key Formula for Taylor Series
Here’s the standard formula: \( f(x) = f(a) + f'(a)\dfrac{(x-a)}{1!} + f''(a)\dfrac{(x-a)^2}{2!} + \ldots + f^{(n)}(a)\dfrac{(x-a)^n}{n!} + \ldots \)
Each term is built from the function’s nth derivative at the point a, multiplied by a power of (x-a), and divided by n! (factorial).
Step-by-Step Illustration
- Choose the function \( f(x) \) you want to expand (e.g., \( f(x) = e^x \) at \( a = 0 \)).
Identify the point \( a \) about which to expand.
- Compute derivatives at \( a \):
\( f'(x), f''(x), f'''(x), \ldots \) and substitute \( x=a \).
- Plug values into the formula:
Write out several terms, increasing the power of (x-a) and factorial in the denominator each time.
- Add the terms to build the series. Stop at enough terms for the desired accuracy.
Examples of Taylor Series for Common Functions
| Function | Taylor Series about a=0 (Maclaurin) |
|---|---|
| \( e^x \) | \( 1 + x + \dfrac{x^2}{2!} + \dfrac{x^3}{3!} + \cdots \) |
| \( \sin x \) | \( x - \dfrac{x^3}{3!} + \dfrac{x^5}{5!} - \cdots \) |
| \( \cos x \) | \( 1 - \dfrac{x^2}{2!} + \dfrac{x^4}{4!} - \cdots \) |
| \( \ln(1+x) \) | \( x - \dfrac{x^2}{2} + \dfrac{x^3}{3} - \dfrac{x^4}{4} + \cdots \) (|x|<1) |
Difference Between Taylor and Maclaurin Series
| Taylor Series | Maclaurin Series |
|---|---|
| Expanded at a general point \( a \): \( f(x) = \sum_{n=0}^\infty f^{(n)}(a) \dfrac{(x-a)^n}{n!} \) |
Special case with \( a = 0 \): \( f(x) = \sum_{n=0}^\infty f^{(n)}(0) \dfrac{x^n}{n!} \) |
So, a Maclaurin series is just a Taylor series centered at zero.
Applications of Taylor Series
Taylor series are super useful in many areas:
- Approximating complicated functions with simple polynomials in calculators or computers
- Solving physics and engineering problems, especially when exact solutions are tough
- Computing limits, derivatives, and integrals for difficult expressions
- Fast estimations and error analysis in numerical methods (see application of derivatives)
Error and Convergence in Taylor Series
Adding more terms in the Taylor series generally gives a better approximation, but the accuracy depends on how “smooth” the function is and the value of \( x \). The error is given by the remainder term—often called the Lagrange remainder.
If the remainder becomes small as the number of terms grows, the series is said to converge to the function.
Sample Problem & Solution: Taylor Series Expansion
Expand \( e^x \) up to the fourth term about \( x = 0 \):
1. Function: \( f(x) = e^x \)2. All its derivatives are \( e^x \); at \( x = 0 \), each derivative is 1
3. Substitute into the formula:
\( f(0) = 1 \)
\( f'(0) = 1 \)
\( f''(0) = 1 \)
\( f'''(0) = 1 \)
4. Series up to third derivative:
\( e^x = 1 + x + \dfrac{x^2}{2!} + \dfrac{x^3}{3!} + \cdots \)
5. Final Answer: \( e^x \approx 1 + x + \dfrac{x^2}{2} + \dfrac{x^3}{6} \)
Frequent Errors and Misunderstandings
- Forgetting to divide by n! for each term’s coefficient
- Misplacing the point a (expanding at the wrong value)
- Using too few terms for wide intervals, causing big error
Relation to Other Concepts
The idea of Taylor Series is closely connected to other calculus topics, such as differentiation rules and power series. Learning Taylor series gives you a foundation for understanding advanced concepts in mathematics, engineering, and science.
Try These Yourself
- Find the Maclaurin series for \( \cos x \) up to fourth degree.
- Approximate \( \ln(1.2) \) using the Taylor series of \( \ln(1+x) \) at \( x=0 \) up to \( x^2 \).
- Write the Taylor polynomial of degree 3 for \( \sin x \) at \( x = 0 \).
Classroom Tip
A quick way to remember Taylor series: each term’s power and factorial is the same as the derivative’s order. Vedantu teachers often use color-coded tables to help students connect terms visually and avoid mistakes during exam revisions.
Wrapping It All Up
We explored Taylor Series—from the definition and formula to solved examples, mistakes, and how it links to other topics. Keep practicing Taylor expansions for various functions using Vedantu’s expert resources. This will build strong fundamentals for JEE, CBSE exams, and higher education.
Related Resources and Calculators
- Power Series – Understand the broader family of infinite expansions in math.
- Binomial Theorem – See how binomial expansions are a special kind of series.
FAQs on Taylor Series: Formula, Expansion & Examples
1. What is the Taylor Series in Maths?
The Taylor Series is a powerful tool in mathematics used to approximate the value of a function at a specific point using its derivatives at another point. It represents a function as an infinite sum of terms, each involving a derivative of the function and a power of the difference between the point of evaluation and the point around which the series is expanded. This allows us to approximate complex functions using simpler polynomial expressions.
2. What is the general formula for Taylor Series expansion?
The general formula for the Taylor series expansion of a function f(x) around a point a is:
f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)²/2! + f'''(a)(x-a)³/3! + ...
Where:
- f(a) is the value of the function at x = a
- f'(a), f''(a), f'''(a), ... are the first, second, third, and higher-order derivatives of the function evaluated at x = a
- n! denotes the factorial of n
3. How is the Taylor Series of ex derived?
The Taylor series for ex is derived by applying the general Taylor series formula. Since all derivatives of ex are ex, and e0 = 1, the series simplifies to:
ex = 1 + x + x²/2! + x³/3! + x⁴/4! + ...
This series converges for all real numbers x.
4. What's the difference between Taylor and Maclaurin Series?
A Maclaurin series is a special case of the Taylor series where the point of expansion is a = 0. Therefore, a Maclaurin series expands a function around the origin. The Taylor series, on the other hand, expands the function around any point a.
5. Where are Taylor Series used in real life or engineering?
Taylor series have numerous applications:
- Approximating functions: Solving complex equations where finding an exact solution is difficult.
- Numerical analysis: Calculating values of functions where direct computation might be challenging.
- Physics and engineering: Modeling systems that are governed by non-linear equations, simplifying calculations.
- Signal processing: Analyzing and manipulating signals in various applications.
- Computer graphics: Smoothing curves and creating realistic images.
6. How do you estimate the error or remainder in a Taylor expansion?
The error in a Taylor expansion is typically estimated using the remainder term, often expressed in Lagrange form. This term involves the next higher-order derivative of the function, evaluated at some point between a and x. The magnitude of this derivative gives an upper bound on the error.
7. How do you decide if a Taylor Series converges for a function?
Convergence of a Taylor series depends on the function and the point of expansion. Various tests can be employed, including the ratio test or root test, to determine the radius of convergence – the range of x values for which the series converges. Outside this radius, the series may diverge.
8. What are the first four terms of the Taylor series for sin(x) when a = 0?
The first four terms of the Maclaurin series (Taylor series with a=0) for sin(x) are:
sin(x) ≈ x - x³/3! + x⁵/5! - x⁷/7!
9. What are higher-order Taylor polynomials used for in numerical analysis?
Higher-order Taylor polynomials provide more accurate approximations of functions. In numerical analysis, they are used in methods like numerical integration and solving differential equations. The higher the order, the better the approximation, although at the cost of increased computational complexity.
10. Can Taylor series be used for non-differentiable functions?
No, the Taylor series requires the function to be infinitely differentiable at the point of expansion. If a function is not differentiable at a point, it cannot be represented by a Taylor series around that point.
11. How to find the interval of convergence of a Taylor series?
The interval of convergence of a Taylor series can be found using various convergence tests like the ratio test or the root test. These tests examine the limit of the ratio (or root) of consecutive terms as the number of terms approaches infinity. The result determines the radius of convergence, and then checking the endpoints will help determine the interval.
12. What is the Taylor series for ln(1+x)?
The Taylor series for ln(1+x) around a=0 is given by:
ln(1+x) = x - x²/2 + x³/3 - x⁴/4 + ...
This series converges for -1 < x ≤ 1.
