Sequences, Series and Power Series

A sequence is an ordered list \(\lbrace a_n\rbrace\), where (in general) \(n=1, 2, 3, \ldots\), having a first element but no last element. In this course, every term in the sequence will be a real number.

Definition 1: Terms for describing sequences
(a) A sequence \(\lbrace a_n\rbrace\) is said to be bounded below by \(L\) if \(a_n\geq L\) for every positive integer \(n\). Then \(L\) is said to be a lower bound for \(\lbrace a_n\rbrace\). Bounded from above and upper bound are defined similarly, and if the sequence is bounded from both above and below, it is said to be bounded.
(b) The sequence \(\lbrace a_n\rbrace\) is positive if it is bounded below by \(0\), and negative if \(0\) is an upper bound.
(c ) The sequence \(\lbrace a_n\rbrace\) is increasing if \(a_{n+1}\geq a_n\) for every positive integer \(n\), and decreasing if \(a_{n+1}\leq a_n\). The sequence is called monotonic if it is either increasing or decreasing.
(d) The sequence \(\lbrace a_n\rbrace\) is alternating if \(a_na_{n+1}<0\) for all positive integers \(n\).

The most important question with sequences is usually whether they converge or not.

Definition 2
We say that a sequence \(\lbrace a_n\rbrace\) converges to a number \(L\) if for every \(\varepsilon>0\) there exists an integer \(N\) (which may depend on \(\varepsilon\)) such that if \(n\geq N\), then \(|a_n-L|<\varepsilon\). We use the notation \(\lim_{n\rightarrow \infty}a_n=L\).

Theorem 1
If \(\lbrace a_n\rbrace\) converges, then \(\lbrace a_n\rbrace\) is bounded.

Theorem 2
If \(\lbrace a_n\rbrace\) is increasing and bounded above, it converges. If it is not bounded above, it diverges to \(\infty\). A similar result holds if it is decreasing.

Definition: Infinite Series
Given a sequence of real numbers \(a_1, a_2, \ldots\), we define the infinite series \(\sum_{n=1}^{\infty} a_n\) to be the limit of partial sums \(\sum_{n=1}^{N} a_n\):

\[ \sum_{n=1}^{\infty} a_n = \lim_{N \to \infty} \sum_{n=1}^{N} a_n. \]
One doesn't get far working with series without the following theorems. Their proofs rely mostly on the definition of infinite series, and a proper understanding of limits of sequences.

Theorem 4
If \( \sum_{n=1}^{\infty} a_n\) converges, then \( \lim_{n \to \infty} a_n = 0 \).

Theorem 5
Given any integer \(N \ge 1\), we have that \(\sum_{n=1}^{\infty} a_n\) converges if and only if \( \sum_{n=N}^{\infty} a_n \) converges.

Theorem 6
If all terms \(a_n\) are nonnegative, then \(\sum_{n=1}^{\infty} a_n\) either converges, or diverges to \(+\infty\).

Theorem 7
Suppose that \(\sum_{n=1}^{\infty} a_n\) and \(\sum_{n=1}^{\infty} b_n\) both converge.
i) Given any constant \(c\), we have that \(\sum_{n=1}^{\infty} (c a_n + b_n) = c \sum_{n=1}^{\infty} a_n + \sum_{n=1}^{\infty} b_n\).
ii) If \(a_n \le b_n\) for all \(n\), then \(\sum_{n=1}^{\infty} a_n \le \sum_{n=1}^{\infty} b_n \).

By Theorem 6 we know that positive series either converge or diverge to \(\infty\). To understand which of these alternatives holds true, is often more interesting than the actual value of the series. There has therefore been developed various methods to test for convergence. Below we assume that all terms of the series are nonnegative.

Theorem 8: The Integral Test
Suppose that \(a_n = f(n)\) for some positive, continuous and decreasing function on the interval \([N, \infty)\) for some positive integer \(N\). Then \[ \sum_{n=1}^{\infty} a_n < \infty \quad \mbox{if and only if} \quad \int_N^{\infty} f(t) \, dt < \infty. \] Theorem 9: A Comparison Test
Suppose that there exists a positive constant \(K\) such that, for all \(n\), \(0\le a_n \le K b_n\).
(a) If the series \(\sum_{n=1}^{\infty} b_n\) converges, then \(\sum_{n=1}^{\infty} a_n\) converges.
(b) As a consequence of (a), we also get that if \(\sum_{n=1}^{\infty} a_n = \infty\), then \(\sum_{n=1}^{\infty} b_n = \infty\).

Theorem 10: A Limit Comparison Test
Suppose that \(\lim_{n \to \infty} a_n/b_n = L\)
(a) If \(L < \infty\) and \(\sum_{n=1}^{\infty} b_n < \infty\), then \(\sum_{n=1}^{\infty} a_n < \infty\).
(b) If \(L>0\), and \(\sum_{n=1}^{\infty} b_n = \infty\), then \(\sum_{n=1}^{\infty} a_n = \infty\).

Theorem 11: The Ratio Test
Suppose that \(\rho = \lim_{n \to \infty} \frac{a_{n+1}}{a_n}\) exists or is \(+\infty\).
(a) If \(0 \le \rho < 1\) then \(\sum_{n=1}^{\infty} a_n\) converges.
(b) If \(1 < \rho \le \infty\), the \(\sum_{n=1}^{\infty} a_n\) diverges to \(+\infty\).

Theorem 12: The Root Test
Suppose that \(\sigma = \lim_{n \to \infty} (a_n)^{1/n} \) exists or is \(+\infty\).
(a) If \(0 \le \sigma < 1\), then \(\sum_{n=1}^{\infty} a_n < \infty\).
(b) If \(1 < \sigma \le \infty \), then \(\sum_{n=1}^{\infty} a_n = \infty\).

Remark: Both the root test and the ratio test are inconclusive when the limit (\(\rho\) or \(\sigma\)) is \(1\).

Working with series where not all terms are positive is not as straightforward. One usually starts by checking convergence of the series where one replaces each term with its absolute value.

Definition 5 and 6
Suppose that \(\sum_{n=1}^{\infty} a_n \) converges.
i) If \(\sum_{n=1}^{\infty} |a_n| \) also converges, we say that the series is absolutely convergent.
ii) If \(\sum_{n=1}^{\infty} |a_n| \) diverges to \(+\infty\), we say that the series is conditionally convergent.

The most interesting conditionally convergent series are those that are alternating.
Theorem 14 and 15
Suppose that \(\{a_n\}_n\) is a sequence such that
i) \(a_n\) and \(a_{n+1}\) have opposite signs,
ii) \(|a_{n+1}|\le |a_n| \),
iii) \(\lim_{n\to \infty} a_n = 0 \).
Then \(\sum_{n=1}^{\infty} a_n\) converges, and for each \(N \ge 1\), we have that \( \left| \sum_{n=N}^{\infty} a_n \right| \le |a_N| \).

Definition 7
A series of the form \(\sum_{n=0}^{\infty} a_n (x-c)^n\) is called a power series.

A natural question is: for which \(x\) does a power series converge? The most important theorem in this respect is
Theorem 17
For any power series \(\sum_{n=0}^{\infty} a_n (x-c)^n\) one of the following alternatives must hold:
i) the series converges absolutely only for \(x=c\),
ii) the series converges absolutely for every \(x\),
iii) there exists \(R>0\) such that we have absolute convergence for \(|x-c| < R\) and divergence for \(|x-c|>R\).
The slogan form of the theorem is: for any power series there exists a radius of convergence, \(R\), which is either 0, finite or infinite.

Inside its radius of convergence, a power series can be both differentiated and integrated.
Theorem 19
Suppose \(\sum_{n=0}^{\infty} a_n (x-c)^n\) has radius of convergence \(R>0\). Then for \(|x|<R\), the power series, seen as a function of \(x\), is differentiable and integrable with \[ \frac{d}{dx} \left( \sum_{n=0}^{\infty} a_n (x-c)^n \right) = \sum_{n=1}^{\infty} n a_n (x-c)^{n-1}, \] and \[ \int_0^x \left( \sum_{n=0}^{\infty} a_n (t-c)^n \right) \, dt = \sum_{n=0}^{\infty} \frac{a_n}{n+1} (x-c)^{n+1}. \]

Theorem 21
Define \(f(x) = \sum_{n=0}^{\infty} a_n (x-c)^n\) for \(|x|<R\), where \(R>0\) is the radius of convergence. Then \[ a_k = \frac{f^{k}(c)}{k!}. \]

Definition 8: Taylor Series
If \(f\) has derivatives of all orders at the point \(c\), then the series \[ \sum_{k=0}^{\infty} \frac{f^{(k)}(c)}{k!}(x-c)^k \] is called the Taylor series of f about c.

It is an important–but not trivial–fact, that although a function has derivatives of all orders, its Taylor series need not equal the function. Functions which are equal to their Taylor series, are called analytic. Exponential, trigonometric and rational functions, together with their inverses, are all analytic wherever they are defined. To prove that a function is analytic, Taylor's Theorem is often useful.