In real analysis, series play an important role in understanding the convergence properties of sequences and functions. A series is essentially the sum of the terms of a sequence, and the study of series provides the foundation for advanced concepts such as Taylor series, Fourier series, and more. This blog will delve into different types of series, their convergence, and some fundamental tests used to analyze them, all while working through examples to better illustrate these concepts.
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Table of Contents
1. Introduction to Series
– Definition of a Series
– Convergence vs Divergence
– Types of Series
– Examples
2. Convergence Tests
– The n-th Term Test
– The Geometric Series Test
– The Integral Test
– The Comparison Test
– Examples
3. **Power Series and Radius of Convergence
– Definition and Explanation
– Examples of Power Series
4. **Taylor Series
– Definition and Explanation
– Examples of Taylor Series
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1. Introduction to Series
1.1 Definition of a Series
A series is the sum of the terms of a sequence. More formally, given a sequence \( (a_n) \), the corresponding series is written as:
\[
S = \sum_{n=0}^{\infty} a_n = a_0 + a_1 + a_2 + \cdots
\]
We say that the series converges to a value \( S \) if the partial sums:
\[
S_N = \sum_{n=0}^{N} a_n
\]
approach \( S \) as \( N \to \infty \). Otherwise, the series is said to diverge.
1.2 Convergence vs Divergence
– Convergent Series: A series \( \sum a_n \) converges to a sum \( S \) if the sequence of partial sums \( S_N \) converges to \( S \).
– Divergent Series: A series \( \sum a_n \) diverges if the sequence of partial sums \( S_N \) does not converge to a finite number.
1.3 Types of Series
– Geometric Series: A series where each term is obtained by multiplying the previous term by a constant ratio. For example:
\[
\sum_{n=0}^{\infty} r^n \quad \text{(where } |r| < 1\text{)}
\]
– p-Series: A series of the form:
\[
\sum_{n=1}^{\infty} \frac{1}{n^p}
\]
where \( p \) is a constant.
– Alternating Series: A series where the terms alternate in sign. For example:
\[
\sum_{n=1}^{\infty} (-1)^n \cdot a_n
\]
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2. Convergence Tests
Several tests are used to determine the convergence or divergence of series. Let’s explore the most common ones.
2.1 The n-th Term Test
The n-th Term Test (also known as the Test for Divergence) states that if:
\[
\lim_{n \to \infty} a_n \neq 0
\]
or if the limit does not exist, then the series \( \sum a_n \) diverges. However, if \( \lim_{n \to \infty} a_n = 0 \), this test is inconclusive (the series might still converge or diverge).
Example 1: n-th Term Test
Consider the series:
\[
\sum_{n=1}^{\infty} \frac{1}{n}
\]
Here, the terms are \( a_n = \frac{1}{n} \). Let’s compute the limit of \( a_n \) as \( n \to \infty \):
\[
\lim_{n \to \infty} \frac{1}{n} = 0
\]
Although the limit is zero, the series diverges, as we know from the harmonic series.
—
2.2 The Geometric Series Test
A geometric series is of the form:
\[
\sum_{n=0}^{\infty} r^n
\]
The geometric series converges if and only if \( |r| < 1 \), and the sum of the series is given by:
\[
S = \frac{1}{1 – r}
\]
If \( |r| \geq 1 \), the series diverges.
Example 2: Geometric Series Test
Consider the series:
\[
\sum_{n=0}^{\infty} \left( \frac{1}{2} \right)^n
\]
Here, \( r = \frac{1}{2} \). Since \( |r| = \frac{1}{2} < 1 \), the series converges, and the sum is:
\[
S = \frac{1}{1 – \frac{1}{2}} = 2
\]
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2.3 The Integral Test
The Integral Test is used when the terms of the series are positive, continuous, and decreasing for sufficiently large \( n \). It states that if the function \( f(x) \) is continuous, positive, and decreasing, and if \( f(n) = a_n \), then:
\[
\sum_{n=1}^{\infty} a_n \text{ converges if and only if } \int_1^\infty f(x) \, dx \text{ converges.}
\]
Example 3: Integral Test
Consider the series:
\[
\sum_{n=1}^{\infty} \frac{1}{n^2}
\]
We can use the function \( f(x) = \frac{1}{x^2} \) to apply the Integral Test. First, compute the improper integral:
\[
\int_1^\infty \frac{1}{x^2} \, dx = \left[ -\frac{1}{x} \right]_1^\infty = 1
\]
Since the integral converges, the series also converges.
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2.4 The Comparison Test
The Comparison Test involves comparing the given series to another series whose convergence is already known. Specifically:
– If \( 0 \leq a_n \leq b_n \) for all \( n \) and \( \sum b_n \) converges, then \( \sum a_n \) converges.
– If \( 0 \leq b_n \leq a_n \) for all \( n \) and \( \sum b_n \) diverges, then \( \sum a_n \) diverges.
Example 4: Comparison Test
Consider the series:
\[
\sum_{n=1}^{\infty} \frac{1}{n^{3/2}}
\]
We compare it to the p-series \( \sum_{n=1}^{\infty} \frac{1}{n^{3/2}} \), which converges because \( p > 1 \). Therefore, by the comparison test, the given series also converges.
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3. Power Series and Radius of Convergence
3.1 Definition and Explanation
A power series is a series of the form:
\[
\sum_{n=0}^{\infty} a_n (x – c)^n
\]
where \( c \) is the center of the series, and \( a_n \) are the coefficients. The radius of convergence \( R \) determines the interval within which the series converges. It can be found using the root test or ratio test.
3.2 Examples of Power Series
Example 1: Power Series for \( e^x \)
The function \( e^x \) can be written as a power series:
\[
e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}
\]
This series converges for all real \( x \), so its radius of convergence is infinite.
4. Taylor Series
4.1 Definition and Explanation
A Taylor series of a function \( f(x) \) around a point \( x = a \) is given by:
\[
f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x – a)^n
\]
where \( f^{(n)}(a) \) is the \( n \)-th derivative of \( f(x) \) evaluated at \( x = a \). The Taylor series provides a polynomial approximation to the function near the point \( a \).
4.2 Examples of Taylor Series
Example 1: Taylor Series for \( \sin x \) at \( x = 0 \)
The Taylor series expansion of \( \sin x \) at \( x = 0 \) is:
\[
\sin x = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!}
\]
Example 2: Taylor Series for \( e^x \) at \( x = 0 \)
The Taylor series expansion of \( e^x \) at \( x = 0 \) is:
\[
e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}
\]
Both of these series are valid for all real values of \( x \), meaning their radius of convergence is infinite.
Conclusion
Series are a fundamental concept in real analysis, and understanding their convergence behavior is essential for working with infinite sums. We’ve explored a variety of tests and types of series, including geometric series, p-series, and alternating series. The convergence tests, such as the n-th term test, geometric series test , and integral test , help determine whether a series converges or diverges. Moreover, power series and Taylor series provide powerful tools for approximating functions. Through examples, we illustrated the application of these concepts and demonstrated their significance in real analysis.
