Taylor Series Formula Derivation and Solved Examples
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 Expansion in Calculus
1. What is a Taylor series in calculus?
A Taylor series is an infinite series that represents a function as a sum of its derivatives evaluated at a single point. It is written as:
f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x - a)^n
Where:
- f^{(n)}(a) is the nth derivative at x = a
- n! is factorial of n
- a is the center of expansion
2. What is the formula for the Taylor series?
The formula for the Taylor series expansion of a function f(x) about x = a is:
f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x - a)^n
Expanded form:
- f(a)
- + f'(a)(x − a)
- + \( \frac{f''(a)}{2!}(x − a)^2 \)
- + \( \frac{f'''(a)}{3!}(x − a)^3 \) + ...
3. What is the difference between Taylor series and Maclaurin series?
The difference is that a Maclaurin series is a special case of the Taylor series centered at a = 0.
- Taylor series: Expanded about x = a
- Maclaurin series: Expanded about x = 0
f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n
So every Maclaurin series is a Taylor series, but not every Taylor series is a Maclaurin series.
4. How do you find the Taylor series of a function?
To find a Taylor series expansion, compute derivatives at the center point and substitute into the Taylor formula.
Steps:
- Choose the center a.
- Find f(a), f'(a), f''(a), etc.
- Substitute into \( \frac{f^{(n)}(a)}{n!}(x-a)^n \).
- Write the series in summation or expanded form.
All derivatives equal e^x, and at 0 they equal 1.
So the series is \( \sum_{n=0}^{\infty} \frac{x^n}{n!} \).
5. What is the Taylor series of e^x?
The Taylor (Maclaurin) series of e^x is:
e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}
Expanded form:
- 1
- + x
- + \( \frac{x^2}{2!} \)
- + \( \frac{x^3}{3!} \) + ...
6. What is the Taylor series of sin x and cos x?
The Maclaurin series for sine and cosine are:
sin x = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!}
cos x = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!}
Expanded forms:
- sin x = x − x³/3! + x⁵/5! − ...
- cos x = 1 − x²/2! + x⁴/4! − ...
7. What is the remainder term in a Taylor series?
The remainder term measures the error between a function and its Taylor polynomial approximation. The Lagrange form is:
R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}
Where:
- c is between a and x
- n is the degree of the Taylor polynomial
8. When does a Taylor series converge to the function?
A Taylor series converges to the function if the remainder term approaches zero as n → ∞. This typically happens within the radius of convergence.
Key points:
- The series converges inside an interval around a.
- The interval depends on the function.
- Some functions (like e^x, sin x, cos x) converge for all real x.
9. What is a Taylor polynomial?
A Taylor polynomial is a finite-degree approximation of a function obtained by truncating its Taylor series. The nth-degree Taylor polynomial is:
T_n(x) = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!}(x-a)^k
It approximates the function near x = a and is commonly used for numerical estimation and error analysis.
10. Why is the Taylor series important in mathematics?
The Taylor series is important because it allows complex functions to be approximated using polynomials. Key applications include:
- Approximating values numerically
- Solving differential equations
- Analyzing limits and continuity
- Modeling in physics and engineering
