Sequences and Series of Real Numbers

1. Sequences of Real Numbers

A sequence of real numbers is an ordered list of real numbers. We can think of it as a function whose domain is the set of natural numbers ($\mathbb{N} = \{1, 2, 3, \dots\}$) and whose range is a subset of the real numbers ($\mathbb{R}$).

We typically denote a sequence by $\{a_n\}_{n=1}^\infty$ or simply $\{a_n\}$, where $a_n$ is the $n$-th term of the sequence. For example, the sequence of even positive integers can be written as $\{2, 4, 6, 8, \dots\}$, or more formally as $\{a_n\}_{n=1}^\infty$ where $a_n = 2n$ for each $n \in \mathbb{N}$.

1.1. Convergence of a Sequence

The most important concept when studying sequences is convergence. A sequence $\{a_n\}$ is said to converge to a real number $L$ if, for every positive number $\epsilon$ (no matter how small), there exists a natural number $N$ such that for all $n \ge N$, the absolute difference between $a_n$ and $L$ is less than $\epsilon$. In symbols:

A sequence $\{a_n\}$ converges to $L$ if $\forall \epsilon > 0, \exists N \in \mathbb{N}$ such that $n \ge N \implies |a_n - L| < \epsilon$.

If such a limit $L$ exists, we say the sequence converges, and we write $\lim_{n \to \infty} a_n = L$. The number $L$ is called the limit of the sequence. If a sequence does not converge, it is said to diverge.

Example: Consider the sequence $\{a_n\}$ where $a_n = \frac{1}{n}$. We claim that $\lim_{n \to \infty} \frac{1}{n} = 0$. To prove this, let $\epsilon > 0$ be given. We need to find an $N$ such that if $n \ge N$, then $|\frac{1}{n} - 0| < \epsilon$. This inequality simplifies to $\frac{1}{n} < \epsilon$, which is equivalent to $n > \frac{1}{\epsilon}$. So, if we choose $N$ to be any integer greater than $\frac{1}{\epsilon}$ (for instance, $N = \lfloor \frac{1}{\epsilon} \rfloor + 1$), then for all $n \ge N$, we will have $n > \frac{1}{\epsilon}$, which implies $\frac{1}{n} < \epsilon$, and thus $|\frac{1}{n} - 0| < \epsilon$. Hence, the sequence converges to 0.

Intuition: The definition means that the terms of the sequence get arbitrarily close to the limit $L$ as $n$ becomes large. For any small "tolerance" $\epsilon$ you choose, you can always find a point in the sequence (from term $N$ onwards) where all subsequent terms are within that tolerance of $L$.

1.2. Properties of Convergent Sequences

If $\{a_n\}$ converges to $L$ and $\{b_n\}$ converges to $M$, then:

• The sum sequence $\{a_n + b_n\}$ converges to $L+M$.
• The difference sequence $\{a_n - b_n\}$ converges to $L-M$.
• The product sequence $\{a_n b_n\}$ converges to $LM$.
• If $M \ne 0$, the quotient sequence $\{\frac{a_n}{b_n}\}$ converges to $\frac{L}{M}$.
• For any constant $c$, the sequence $\{c a_n\}$ converges to $cL$.

These properties are extremely useful because they allow us to find the limits of complex sequences by breaking them down into simpler ones.

1.3. Monotone Sequences

A sequence $\{a_n\}$ is called:

• Increasing if $a_n \le a_{n+1}$ for all $n$.
• Decreasing if $a_n \ge a_{n+1}$ for all $n$.
• Monotone if it is either increasing or decreasing.

A sequence $\{a_n\}$ is called:

• Strictly increasing if $a_n < a_{n+1}$ for all $n$.
• Strictly decreasing if $a_n > a_{n+1}$ for all $n$.

1.4. Bounded Sequences

A sequence $\{a_n\}$ is called:

• Bounded above if there exists a real number $U$ such that $a_n \le U$ for all $n$.
• Bounded below if there exists a real number $L$ such that $a_n \ge L$ for all $n$.
• Bounded if it is both bounded above and bounded below.

1.5. Monotone Convergence Theorem

This is a fundamental theorem in real analysis. It states that:

• A monotone increasing sequence that is bounded above converges.
• A monotone decreasing sequence that is bounded below converges.

Furthermore, the limit of such a sequence is the supremum (least upper bound) if it's increasing and bounded above, or the infimum (greatest lower bound) if it's decreasing and bounded below.

Example: Consider the sequence $a_n = \frac{n}{n+1}$.

Is it increasing? $a_{n+1} = \frac{n+1}{n+2}$. We compare $a_n$ and $a_{n+1}$.

$a_{n+1} - a_n = \frac{n+1}{n+2} - \frac{n}{n+1} = \frac{(n+1)^2 - n(n+2)}{(n+2)(n+1)} = \frac{n^2+2n+1 - (n^2+2n)}{(n+1)(n+2)} = \frac{1}{(n+1)(n+2)}$. Since this is always positive for $n \in \mathbb{N}$, $a_{n+1} > a_n$, so the sequence is strictly increasing.

Is it bounded above? As $n$ gets large, $a_n = \frac{n}{n+1}$ approaches 1. We can see that $a_n = \frac{n+1-1}{n+1} = 1 - \frac{1}{n+1}$. Since $\frac{1}{n+1} > 0$, $a_n$ is always less than 1. So, the sequence is bounded above by 1.

Since the sequence is monotone increasing and bounded above, by the Monotone Convergence Theorem, it converges. In fact, we know $\lim_{n \to \infty} (1 - \frac{1}{n+1}) = 1 - 0 = 1$. The limit is 1, which is the supremum of the sequence terms.

1.6. Cauchy Sequences

Another important characterization of convergent sequences is the Cauchy criterion. A sequence $\{a_n\}$ is a Cauchy sequence if, for every $\epsilon > 0$, there exists a natural number $N$ such that for all $m, n \ge N$, we have $|a_m - a_n| < \epsilon$.

Theorem: A sequence of real numbers converges if and only if it is a Cauchy sequence.

This theorem is powerful because it allows us to determine convergence without knowing the limit beforehand. It focuses on the internal behavior of the sequence terms.

1.7. Divergent Sequences

Sequences that do not converge are called divergent. Common ways for a sequence to diverge include:

• Diverging to infinity: The terms become arbitrarily large (positive or negative). For example, $\{n^2\}$ diverges to $+\infty$. We write $\lim_{n \to \infty} n^2 = \infty$.
• Oscillating: The terms do not approach a single value. For example, $\{(-1)^n\}$ alternates between -1 and 1 and does not converge.

Formal definition for divergence to infinity: A sequence $\{a_n\}$ diverges to $\infty$ if $\forall M > 0, \exists N \in \mathbb{N}$ such that $n \ge N \implies a_n > M$. Similarly for divergence to $-\infty$.

2. Series of Real Numbers

A series is the sum of the terms of a sequence. If $\{a_n\}$ is a sequence of real numbers, the series associated with it is denoted by $\sum_{n=1}^\infty a_n$. This notation represents the sum $a_1 + a_2 + a_3 + \dots$

To define the sum of an infinite series, we use the concept of partial sums. Let $\{a_n\}$ be a sequence. Define the sequence of partial sums $\{S_k\}$ as:

• $S_1 = a_1$
• $S_2 = a_1 + a_2$
• $S_3 = a_1 + a_2 + a_3$
• ...
• $S_k = \sum_{n=1}^k a_n = a_1 + a_2 + \dots + a_k$

2.1. Convergence of a Series

A series $\sum_{n=1}^\infty a_n$ is said to converge if the sequence of its partial sums $\{S_k\}$ converges to a real number $S$. This limit $S$ is called the sum of the series, and we write $\sum_{n=1}^\infty a_n = S$. If the sequence of partial sums $\{S_k\}$ diverges, then the series $\sum_{n=1}^\infty a_n$ is said to diverge.

Example: Geometric Series

Consider the series $\sum_{n=0}^\infty r^n = 1 + r + r^2 + r^3 + \dots$ where $r$ is a real number.

The $k$-th partial sum is $S_k = 1 + r + r^2 + \dots + r^{k-1}$.

If $r=1$, $S_k = 1 + 1 + \dots + 1$ ($k$ times) $= k$. The sequence of partial sums is $\{1, 2, 3, \dots\}$, which diverges to $\infty$. So, $\sum_{n=0}^\infty 1^n$ diverges.

If $r \ne 1$, we can use the formula for a finite geometric sum: $S_k = \frac{1 - r^k}{1 - r}$.

Now we examine the limit of $S_k$ as $k \to \infty$.

• If $|r| < 1$, then $\lim_{k \to \infty} r^k = 0$. So, $\lim_{k \to \infty} S_k = \frac{1 - 0}{1 - r} = \frac{1}{1 - r}$. The series converges to $\frac{1}{1-r}$.
• If $|r| > 1$, then $|r^k|$ grows without bound as $k \to \infty$, so $S_k$ diverges. The series diverges.
• If $r = -1$, the partial sums are $1, 0, 1, 0, \dots$, which oscillates and diverges. The series diverges.

Conclusion for Geometric Series: $\sum_{n=0}^\infty r^n$ converges if and only if $|r| < 1$, in which case its sum is $\frac{1}{1-r}$.

2.2. Necessary Condition for Convergence (Test for Divergence)

If a series $\sum_{n=1}^\infty a_n$ converges, then the sequence of its terms must converge to 0. That is, if $\sum_{n=1}^\infty a_n$ converges, then $\lim_{n \to \infty} a_n = 0$.

The contrapositive is also true and is more useful as a test: If $\lim_{n \to \infty} a_n \ne 0$ (or if the limit does not exist), then the series $\sum_{n=1}^\infty a_n$ diverges.

This is called the Test for Divergence.

Example: Harmonic Series

Consider the harmonic series $\sum_{n=1}^\infty \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots$. Here, $a_n = \frac{1}{n}$. We know that $\lim_{n \to \infty} \frac{1}{n} = 0$. The Test for Divergence gives no information.

However, the harmonic series is famously divergent. One way to show this is by grouping terms:

$1 + \frac{1}{2} + (\frac{1}{3} + \frac{1}{4}) + (\frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8}) + \dots$

Notice that:

• $\frac{1}{3} + \frac{1}{4} > \frac{1}{4} + \frac{1}{4} = \frac{2}{4} = \frac{1}{2}$
• $\frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8} > \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} = \frac{4}{8} = \frac{1}{2}$

The sum can be bounded below by $1 + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \dots$, which clearly diverges to infinity. Therefore, the harmonic series diverges.

2.3. Tests for Convergence of Series with Positive Terms

When all terms $a_n$ of a series are positive ($a_n > 0$), the sequence of partial sums $\{S_k\}$ is strictly increasing. By the Monotone Convergence Theorem, this sequence will either converge (if it's bounded above) or diverge to infinity (if it's not bounded above). So, a series of positive terms either converges or diverges to infinity.

a) Integral Test

Suppose $f(x)$ is a positive, continuous, and decreasing function on the interval $[1, \infty)$. Then the series $\sum_{n=1}^\infty f(n)$ converges if and only if the improper integral $\int_1^\infty f(x) dx$ converges.

Example: p-Series

A p-series has the form $\sum_{n=1}^\infty \frac{1}{n^p}$ for some real number $p$. Let $f(x) = \frac{1}{x^p}$. This function is positive, continuous, and decreasing for $x \ge 1$ if $p > 0$. We apply the Integral Test.

Consider the integral $\int_1^\infty \frac{1}{x^p} dx$.

• If $p > 1$: $\int_1^\infty x^{-p} dx = \lim_{b \to \infty} [\frac{x^{-p+1}}{-p+1}]_1^b = \lim_{b \to \infty} (\frac{b^{1-p}}{1-p} - \frac{1}{1-p})$. Since $1-p < 0$, $b^{1-p} \to 0$ as $b \to \infty$. The integral converges to $-\frac{1}{1-p} = \frac{1}{p-1}$.
• If $0 < p \le 1$: $\int_1^\infty x^{-p} dx = \lim_{b \to \infty} [\frac{x^{-p+1}}{-p+1}]_1^b$. Since $-p+1 \ge 0$, $b^{-p+1} \to \infty$ as $b \to \infty$. The integral diverges.

Conclusion: p-Series Test

The p-series $\sum_{n=1}^\infty \frac{1}{n^p}$ converges if $p > 1$ and diverges if $p \le 1$.

(The harmonic series is the case $p=1$, which diverges.)

b) Comparison Test

Let $\sum a_n$ and $\sum b_n$ be series with positive terms.

• If $a_n \le b_n$ for all $n$ (or for all $n \ge N$ for some $N$), and $\sum b_n$ converges, then $\sum a_n$ also converges. (If the "larger" series converges, the "smaller" one must too).
• If $a_n \ge b_n$ for all $n$ (or for all $n \ge N$), and $\sum b_n$ diverges, then $\sum a_n$ also diverges. (If the "smaller" series diverges, the "larger" one must too).

c) Limit Comparison Test

Let $\sum a_n$ and $\sum b_n$ be series with positive terms. If $\lim_{n \to \infty} \frac{a_n}{b_n} = c$, where $c$ is a finite positive number ($0 < c < \infty$), then either both series converge or both series diverge.

This test is often easier to use than the direct Comparison Test when the terms $a_n$ and $b_n$ behave similarly for large $n$. We often choose $b_n$ to be a term from a known series (like a p-series or geometric series) that has the same "dominant" terms as $a_n$.

Example: Determine if $\sum_{n=1}^\infty \frac{n^2 + 3n}{n^4 - n + 1}$ converges.

The terms $a_n = \frac{n^2 + 3n}{n^4 - n + 1}$ are positive for large $n$. For large $n$, the dominant term in the numerator is $n^2$ and in the denominator is $n^4$. So, $a_n$ behaves like $\frac{n^2}{n^4} = \frac{1}{n^2}$.

Let's choose $b_n = \frac{1}{n^2}$. We know $\sum b_n = \sum \frac{1}{n^2}$ is a p-series with $p=2 > 1$, so it converges.

Now, we compute the limit of the ratio:

$\lim_{n \to \infty} \frac{a_n}{b_n} = \lim_{n \to \infty} \frac{\frac{n^2 + 3n}{n^4 - n + 1}}{\frac{1}{n^2}} = \lim_{n \to \infty} \frac{n^2(n^2 + 3n)}{n^4 - n + 1} = \lim_{n \to \infty} \frac{n^4 + 3n^3}{n^4 - n + 1}$

Divide numerator and denominator by $n^4$: $\lim_{n \to \infty} \frac{1 + \frac{3}{n}}{1 - \frac{1}{n^3} + \frac{1}{n^4}} = \frac{1+0}{1-0+0} = 1$.

Since the limit is $c=1$, which is finite and positive, and $\sum b_n$ converges, by the Limit Comparison Test, $\sum a_n$ also converges.

d) Ratio Test

Let $\sum a_n$ be a series with positive terms. Calculate the limit $L = \lim_{n \to \infty} \frac{a_{n+1}}{a_n}$.

• If $L < 1$, the series converges.
• If $L > 1$, the series diverges.
• If $L = 1$, the test is inconclusive.

This test is particularly useful for series involving factorials or exponential terms.

Example: $\sum_{n=1}^\infty \frac{2^n}{n!}$.

$a_n = \frac{2^n}{n!}$. Then $a_{n+1} = \frac{2^{n+1}}{(n+1)!}$.

$L = \lim_{n \to \infty} \frac{a_{n+1}}{a_n} = \lim_{n \to \infty} \frac{\frac{2^{n+1}}{(n+1)!}}{\frac{2^n}{n!}} = \lim_{n \to \infty} \frac{2^{n+1}}{(n+1)!} \cdot \frac{n!}{2^n} = \lim_{n \to \infty} \frac{2 \cdot 2^n}{(n+1) \cdot n!} \cdot \frac{n!}{2^n} = \lim_{n \to \infty} \frac{2}{n+1} = 0$.

Since $L=0 < 1$, the series converges by the Ratio Test.

e) Root Test

Let $\sum a_n$ be a series with positive terms. Calculate the limit $L = \lim_{n \to \infty} \sqrt[n]{a_n}$.

• If $L < 1$, the series converges.
• If $L > 1$, the series diverges.
• If $L = 1$, the test is inconclusive.

This test is often useful when terms involve powers of $n$.

2.4. Alternating Series

An alternating series is a series where the terms alternate in sign. The general form is $\sum_{n=1}^\infty (-1)^{n-1} b_n = b_1 - b_2 + b_3 - b_4 + \dots$ or $\sum_{n=1}^\infty (-1)^n b_n = -b_1 + b_2 - b_3 + b_4 - \dots$, where $b_n > 0$ for all $n$.

Alternating Series Test (Leibniz Test):

An alternating series $\sum_{n=1}^\infty (-1)^{n-1} b_n$ converges if the following two conditions are met:

1. $b_n \ge b_{n+1}$ for all $n$ (or for all $n \ge N$ for some $N$); i.e., the absolute values of the terms are non-increasing.
2. $\lim_{n \to \infty} b_n = 0$.

Absolute Convergence vs. Conditional Convergence

A series $\sum a_n$ is called absolutely convergent if the series of the absolute values, $\sum |a_n|$, converges.

A series $\sum a_n$ is called conditionally convergent if $\sum a_n$ converges, but $\sum |a_n|$ diverges.

Key Theorem: If a series converges absolutely, then it converges.

This means if $\sum |a_n|$ converges, then $\sum a_n$ also converges. The converse is not true.

For alternating series, the Alternating Series Test guarantees convergence of $\sum a_n$. If $\sum |a_n|$ also converges, then the series is absolutely convergent. If $\sum |a_n|$ diverges, then the series is conditionally convergent.

Example: Alternating Harmonic Series

Consider $\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \dots$.

Here $b_n = \frac{1}{n}$.

1. $b_n = \frac{1}{n}$ is decreasing since $\frac{1}{n} > \frac{1}{n+1}$.
2. $\lim_{n \to \infty} b_n = \lim_{n \to \infty} \frac{1}{n} = 0$.

Both conditions of the Alternating Series Test are met, so the series converges.

Now consider the series of absolute values: $\sum_{n=1}^\infty |\frac{(-1)^{n-1}}{n}| = \sum_{n=1}^\infty \frac{1}{n}$, which is the harmonic series. We know this diverges.

Since $\sum a_n$ converges but $\sum |a_n|$ diverges, the alternating harmonic series is conditionally convergent.

2.5. Power Series

A power series is a series of the form $\sum_{n=0}^\infty c_n (x-a)^n = c_0 + c_1(x-a) + c_2(x-a)^2 + \dots$, where $c_n$ are constants and $x$ is a variable. The point $a$ is called the center of the series.

For a given power series, there is a radius of convergence $R$ ($R$ can be 0, a positive number, or $\infty$) and possibly an interval of convergence.

• If $R=0$, the series only converges at $x=a$.
• If $R$ is a positive finite number, the series converges absolutely for $|x-a| < R$, and diverges for $|x-a| > R$. Convergence at the endpoints $x = a-R$ and $x = a+R$ must be tested separately using other convergence tests.
• If $R=\infty$, the series converges absolutely for all real numbers $x$.

The radius of convergence $R$ can often be found using the Ratio Test or Root Test applied to the series of absolute values $\sum |c_n (x-a)^n|$.

Using Ratio Test for Radius of Convergence:

Let $a_n = c_n (x-a)^n$. Then $a_{n+1} = c_{n+1} (x-a)^{n+1}$.

Consider $\lim_{n \to \infty} |\frac{a_{n+1}}{a_n}| = \lim_{n \to \infty} |\frac{c_{n+1}(x-a)^{n+1}}{c_n(x-a)^n}| = |x-a| \lim_{n \to \infty} |\frac{c_{n+1}}{c_n}|$.

For convergence, this limit must be less than 1. If $\lim_{n \to \infty} |\frac{c_{n+1}}{c_n}| = L$, then we need $|x-a| L < 1$, which means $|x-a| < \frac{1}{L}$. So, $R = \frac{1}{L}$ (provided $L \ne 0$ and $L \ne \infty$).

Taylor and Maclaurin Series:

A power series can represent a function. Conversely, under certain conditions, a function can be represented by a power series. If a function $f(x)$ has derivatives of all orders at $x=a$, its Taylor series is given by:

$f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!} (x-a)^n = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \dots$

If $a=0$, this is called the Maclaurin series: $f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!} x^n$.

Many common functions have well-known Maclaurin series, such as:

• $e^x = \sum_{n=0}^\infty \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots$ (Converges for all $x$)
• $\sin x = \sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots$ (Converges for all $x$)
• $\cos x = \sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots$ (Converges for all $x$)
• $\frac{1}{1-x} = \sum_{n=0}^\infty x^n = 1 + x + x^2 + x^3 + \dots$ (Converges for $|x|<1$)

2.6. Rearrangement of Series

For absolutely convergent series, the sum is independent of the order of the terms. You can rearrange the terms in any way, and the series will still converge to the same sum.

However, for conditionally convergent series, the sum depends heavily on the order of the terms. Riemann's rearrangement theorem states that a conditionally convergent series can be rearranged to converge to any real number, or even diverge to $\pm \infty$. This highlights the critical importance of absolute convergence.