16 Sequences and Series of Real Numbers
We have already looked at limits of sequences in a metric space, as a generalization of the limit of a sequence in $\R$. Chapter 4 in the textbook takes a more in-depth look at sequences in $\R$, and it covers series (infinite sums) in detail. Some of the material will be familiar from Calculus II, but there are many new ideas.
We consider sequences of real numbers. To review: Remember that a sequence $\{x_n\}_{n=1}^\infty$ is really a function from $\N$ to $\R$. We say that $\ds\lim_{n\to\infty}x_n=L$ if for every $\eps>0$, there is an $N\in\N$ such that $|x_n-L|<\eps$ for all $n\ge N$. If a sequence has a limit, we say that the sequence converges; a seqence that does not converge is said to diverge. A special case of divergence is divergence to $\pm\infty$. We say $\ds\lim_{n\to\infty}x_n=+\infty$ if for every $M\in\R$, there is an $N\in\N$ such that $x_n>M$ for all $n\ge N$. We say $\ds\lim_{n\to\infty}x_n=-\infty$ if for every $M\in\R$, there is an $N\in\N$ such that $x_n<M$ for all $n\ge N$. It is important to remember that when $\ds\lim_{n\to\infty}x_n=+\infty$ or $\ds\lim_{n\to\infty}x_n=-\infty$, the sequence $\{x_n\}_{n=1}^\infty$ is divergent.
The new material on sequences of real numbers is mainly concerned with two important classes of sequences: monotone sequences and Cauchy sequences.
Definition: A sequence $\{x_n\}_{n=1}^\infty$ is increasing if for all $n,m\in\N$, if $n>m$ then $x_n\ge x_m$. It is decreasing if for all $n,m\in\N$, if $n>m$ then $x_n\le x_m$. And is is monotone if it is either increasing or decreasing. (Note that "increasing" in this definition should probably be called non-decreasing, and "decreasing" should be non-increasing.)
It should be clear that an increasing sequence that is bounded above is convergent. In fact, it converges to $lub\{x_n\,|\,n\in\N\}$. Similarly, a decreasing sequence that is bounded below converges to its greatest lower bound. These facts are stated in this theorem:
Monotone Convergence Theorem: An increasing sequence that is bounded above is convergent. An increasing sequence that is not bounded above diverges to infinity. A decreasing sequence that is bounded below is convergent. A decreasing sequence that is not bounded below diverges to minus infinity.
A Cauchy sequence is one in which the terms of the sequence get arbitrarily close to each other as $n\to\infty$, The major result is that any Cauchy sequence of real numbers is convergent. Note that both this fact and the Monotone Convergence Theorem depend on the completeness of $\R$ in an essential way. For example, it is not true that a Cauchy sequence of rational numbers must converge to a rational number. We also consider a particular type of sequence called a "contraction."
Cauchy Convergence Theorem: Any Cauchy sequence of real numbers is convergent.
Theorem (The Contraction Principle): Let $\{x_n\}_{n=1}^\infty$ be a sequence of real numbers. Suppose that there is a real number $r$ in the range $0<r<1$ such that for all $n$, $|a_{n+2}-a_{n+1}|\le|a_{n+1}-a_n|$. Then the sequence is convergent.
Note by the way that all of these convergence criteria will apply to a sequence $\{x_n\}_{n=1}^\infty$ that "eventually" meets the criteria, that is if there is an $K\in\N$ such that the criterion holds for the subsequence $\{x_n\}_{n=K}^\infty$. This is because $\{x_n\}_{n=1}^\infty$ and $\{x_n\}_{n=K}^\infty$ clearly have the same convergence behavior (since convergence is about what happens to $x_n$ in the long run, which doesn't depend on the first $K$ terms of the sequence). For example, a bounded-above sequence will converge by the Monotone Convergence Sequence if it is consistently increasing after the $K^{\rm th}$ term, no matter what it does for the first $K$ terms.
Closely related to infinite sequences are infinite series. An infinite series is written as an infinite sum such as $\ds\sum_{n=1}^\infty a_n$. To make sense of such a sum, we have to look at the infinite sequence of partial sums of the series.
Definition: The k-th partial sum of the infinite series $\ds\sum_{n=1}^\infty a_n$ is the value of the finite sum $\ds\sum_{n=1}^ka_n$. The infinite series is convergent if the sequence of partial sums, $\ds\Big\{\sum_{n=1}^ka_n\Big\}_{k=1}^\infty$, is convergent; otherwise the series is divergent. For a convergent series, the sum of the series is defined to be the limit of the sequence of partial sums, and we write $\ds\sum_{n=1}^\infty a_n=\lim_{k\to\infty}\sum_{n=1}^ka_n$. Similarly, we write $\ds\sum_{n=1}^\infty a_n = \pm\infty$ when $\ds\lim_{k\to\infty}\sum_{n=1}^ka_n=\pm\infty$.
Much of the theory of infinite sequences carries over to infinite series. For example, suppose that $a_n\ge 0$ for all $n$. In that case, the sequence of partial sums is increasing, so the Monotone Convergence Theorem applies to the sequence of partial sums. This means that the series will either converge or will diverge to infinity, depending on whether or not the sequence of partial sums is bounded above. This fact is one of the reasons that we often consider series of non-negative terms. A non-negative series is one whose terms are all greater than or equal to zero.
Major results about series include a variety of tests for convergence and divergence. Note that many of these tests apply only to non-negative series. However, the $n^{\rm th}$ term test applies to any series. It says that a series cannot converge unless its terms get small. It is usually used as a test for divergence.
Theorem (The $n^{\rm th}$ term test) If the series $\sum_{n=1}^\infty a_n$ converges, then $\lim_{n\to\infty} a_n = 0$. Equivalently if $\ds\lim_{n\to\infty} a_n$ does not exist, or if the limit is some non-zero value, then the series $\sum_{n=1}^\infty$ diverges.
Theorem (Comparison test) Suppose that $\sum_{n=1}^\infty a_n$ and $\sum_{n=1}^\infty b_n$ are non-negative series.
If $a_n\le b_n$ for all $n$ and if $\sum_{n=1}^\infty b_n$ converges, then $\sum_{n=1}^\infty a_n$ also converges.
If $a_n\ge b_n$ for all $n$ and if $\sum_{n=1}^\infty b_n$ diverges, then $\sum_{n=1}^\infty a_n$ also diverges.
Theorem (Ratio test) Suppose that $\sum_{n=1}^\infty a_n$ is a non-negative series, and that $\ds\lim_{n\to\infty}\frac{a_{n+1}}{a_n}=L$ (where $L$ can be either a non-negative number or $+\infty$). If $0\le L<1$, then the series converges. If $L>1$, then the series diverges. (If $L=1$, then the test gives no information about convergence of the series.)
Theorem (Root test) Suppose that $\sum_{n=1}^\infty a_n$ is a non-negative series, and that $\ds\lim_{n\to\infty}\root n \of {a_n}=L$ (where $L$ can be either a non-negative number or $+\infty$). If $0\le L<1$, then the series converges. If $L>1$, then the series diverges. (If $L=1$, then the test gives no information about convergence of the series.)
Infinite series also have a linearity property, similar to the linearity property of the definite integral:
Theorem: If $\sum_{n=1}^\infty a_n$ and $\sum_{n=1}^\infty b_n$ are convergent series, then the series $\sum_{n=1}^\infty (a_n+b_n)$ also converges, and $$\sum_{n=1}^\infty (a_n+b_n) = \sum_{n=1}^\infty a_n + \sum_{n=1}^\infty b_n$$ If $\sum_{n=1}^\infty a_n$ is a convergent series and $r\in\R$ then $\sum_{n=1}^\infty ra_n$ also converges, and $$\sum_{n=1}^\infty ra_n = r\cdot \sum_{n=1}^\infty a_n$$
For series that can have both positive and negative terms, we can distinguish between "absolute" convergence and "conditional" convergence. It can be shown that a series that is absolutely convergent is also convergent. And we get one more test for convergence that applies to alternating series, in which the terms alternate between positive and negative.
Definition: Let $\sum_{n=1}^\infty a_n$ be a series. The series is said to be absolutely convergent if the series $\sum_{n=1}^\infty |a_n|$ is convergent. A series that is convergent but not absolutely convergent is said to be conditionally convergent. That is, $\sum_{n=1}^\infty a_n$ is conditionally convergent if it converges but $\sum_{n=1}^\infty|a_n|$ diverges.
Theorem: If the series $\sum_{n=1}^\infty a_n$ converges absolutely, then it converges.
Theorem (Alternating Series Test) Let $\{a_n\}_{n=1}^\infty$ be a decreasing sequence of non-negative terms, such that $\ds\lim_{n\to\infty}a_n=0$. Then the series $\sum_{n=1}^\infty (-1)^na_n$ converges.
The most typical example of a conditionally convergent series is $\sum_{n=1}^\infty\frac{(-1)^n}{n}$. This series converges by the Alternating Series Test, but the series of absolute values is the well-known harmonic series, $\sum_{n=1}^\infty\frac1n$, which diverges.
It is worth noting that the Ratio and Root tests can be adapted for series that can include negative as well as non-negative terms, using absolute convergence:
Theorem (Ratio test) Let $\sum_{n=1}^\infty a_n$ be any series, and suppose that $\ds\lim_{n\to\infty}\frac{|a_{n+1}|}{|a_n|}=L$ (where $L$ can be either a non-negative number or $+\infty$.) If $0\le L<1$, then the series converges absolutely. If $L>1$, then the series diverges. (If $L=1$, then the test gives no information about convergence of the series.)
Theorem (Root test) Let $\sum_{n=1}^\infty a_n$ be any series, and suppose that $\ds\lim_{n\to\infty}\root n \of {|a_n|}=L$ (where $L$ can be either a non-negative number or $+\infty$.) If $0\le L<1$, then the series converges absolutely. If $L>1$, then the series diverges. (If $L=1$, then the test gives no information about convergence of the series.)