Specially Series Expansion Based Problems

Series expansions are a fundamental tool in calculus and analysis, used to approximate functions using a sum of simpler terms. They are particularly useful for solving problems involving limits, differentiation, and integration when the functions involved are not easily manageable. In this article, we will delve into the topic of series expansion based problems, focusing on their applications in solving limits.

Understanding Series Expansions

Before we tackle specific problems, it's important to understand the different types of series expansions that are commonly used:

1. Taylor Series: Expands a function into an infinite sum of terms calculated from the values of its derivatives at a single point.
2. Maclaurin Series: A special case of the Taylor series, where the expansion is around zero.
3. Laurent Series: Similar to the Taylor series but allows for negative powers and is used for functions with singularities.
4. Fourier Series: Represents a function as a sum of sines and cosines, useful for periodic functions.

Key Points of Series Expansions

Point	Description
Convergence	A series must converge to a finite value for the expansion to be valid.
Interval of Convergence	The range of input values for which the series converges to the function.
Remainder Term	The difference between the function and the finite sum of its series expansion.
Differentiability	The function must be differentiable up to a certain order for a Taylor or Maclaurin series.
Singularities	For functions with singularities, a Laurent series may be more appropriate.

Formulas for Series Expansions

Taylor Series

The Taylor series expansion of a function $f(x)$ about a point $a$ is given by:

$$ f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \ldots $$

Maclaurin Series

The Maclaurin series is a Taylor series centered at $a=0$:

$$ f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \ldots $$

Fourier Series

For a periodic function $f(x)$ with period $2\pi$, the Fourier series is:

$$ f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left( a_n \cos(nx) + b_n \sin(nx) \right) $$

where

$$ a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos(nx) \, dx, \quad b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin(nx) \, dx $$

Examples of Series Expansion Based Problems

Example 1: Maclaurin Series for Exponential Function

Find the Maclaurin series for $e^x$.

Solution:

The derivatives of $e^x$ are all $e^x$, and at $x=0$, all derivatives are equal to 1. Thus, the Maclaurin series is:

$$ e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots = \sum_{n=0}^{\infty} \frac{x^n}{n!} $$

Example 2: Limit Using Series Expansion

Evaluate the limit $\lim_{x \to 0} \frac{\sin(x) - x}{x^3}$.

Solution:

Using the Maclaurin series for $\sin(x)$:

$$ \sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \ldots $$

Substitute this into the limit:

$$ \lim_{x \to 0} \frac{\sin(x) - x}{x^3} = \lim_{x \to 0} \frac{x - \frac{x^3}{3!} + \frac{x^5}{5!} - \ldots - x}{x^3} = \lim_{x \to 0} \frac{- \frac{x^3}{3!} + \frac{x^5}{5!} - \ldots}{x^3} $$

As $x$ approaches 0, all terms with higher powers of $x$ vanish, and we are left with:

$$ \lim_{x \to 0} \frac{- \frac{x^3}{3!}}{x^3} = -\frac{1}{3!} = -\frac{1}{6} $$

Example 3: Taylor Series for $\ln(1+x)$

Find the Taylor series for $\ln(1+x)$ about $x=0$.

Solution:

The derivatives of $\ln(1+x)$ at $x=0$ are:

$$ f(x) = \ln(1+x), \quad f'(x) = \frac{1}{1+x}, \quad f''(x) = -\frac{1}{(1+x)^2}, \ldots $$

Evaluating at $x=0$ gives $f(0)=0, f'(0)=1, f''(0)=-1, \ldots$. The series is:

$$ \ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \ldots = \sum_{n=1}^{\infty} (-1)^{n+1}\frac{x^n}{n} $$

Conclusion

Series expansions are powerful tools for approximating functions and solving complex mathematical problems, particularly those involving limits. By understanding the different types of series and their properties, students can tackle a wide range of problems in calculus and analysis. It is important to remember the conditions under which these series are valid and to be able to determine the interval of convergence for a given series.