**Weekly schedule:**

- Semesteruke 12

- Semesteruke 13

**Lectures:**Lectures 29, 30, 31, 32

**Textbook:**Sections 9.1-9.7

**Videos:**Sequences, Series and Power Series Videos

**Examples:**Sequences, Series and Power Series examples

**Maple:**

-:Convergence tests

-:Power series

-:Problem 9.3.28

-:Problem 9.5.12

**Exercises:** Sequences, Series and Power Series exercises

**Project problems:** Sequences, Series and Power Series project problems

**Maple TA test:** Maple TA

# Sequences, Series and Power Series

A series is a sum with an infinite number of terms. As with any concept related to infinity, the definition of a series involves taking a limit. The most interesting case is when each term in the series is a function; the resulting series is then a function as well. In fact, it can be shown that all the functions we have been working with so far – exponential, trigonometric and rational functions, together with their inverses – can be (locally) represented as an "infinite polynomial". This will extend our understanding of these functions even further.

## Topics:

Sequences and Convergence

Sequences and Convergence

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.

**Relevant parts of the book:**Section 9.1

Infinite Series

Infinite Series

**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 \(\sum_{n=1}^{\infty} a_n \le \sum_{n=1}^{\infty} b_n \).

**Relevant parts of the book:**Section 9.2

**Relevant videos:**

Følger og rekker

Hva skal vi med rekker?

**Relevant examples:**The converse of Theorem 4 does not hold

Convergence Tests for Positive Series

Convergence Tests for Positive Series

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 \(1\).

**Relevant parts of the book:**Section 9.3

**Relevant videos:**

En oppgave med integraltesten

Aug 1998 - oppgave 2

Aug 1999 - oppgave 5

**Maple:**

-:Convergence tests

-:Problem 9.3.28

**Pencasts:**

- Exercise 9-3-1: the integral test (2:21)

- Exercise 9-3-11

- Exercise 9-3-18

- Exercise 9-3-24: estimating from below (1:18)

- Exercise 9-3-39: a more complicated series, part 1, using the root test (1:33)

- Exercise 9-3-40: a more complicated series, Part 2, using the ratio test (2:15)

**Relevant examples:**Using the root test

Absolute and Conditional Convergence

Absolute and Conditional Convergence

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 \( \left| \sum_{n=N}^{\infty} a_n \right| \le |a_N| \).

Power Series

Power Series

**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
\[
\int_0^x \left( \sum_{n=0}^{\infty} a_n (t-c)^n \right) \, dt = \sum_{n=0}^{\infty} \frac{a_n}{n+1} (x-c)^{n+1}.
\]

**Relevant parts of the book:**Section 9.5

**Relevant videos:**

Potensrekker

Aug 1998 - oppgave 6

**Maple:**

-:Power series

-:Problem 9.5.12

**Relevant examples:**Finding the sum of a power series by differentiation

Taylor Series

Taylor Series

**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.

**Relevant parts of the book:**Sections 9.6, 9.7

**Relevant videos:**

Å representere en gitt funksjon ved potensrekke

**Pencasts:**Exercise 9-7-15

**Weekly schedule:**

- Semesteruke 12

- Semesteruke 13

**Lectures:**Lectures 29, 30, 31, 32

**Textbook:**Sections 9.1-9.7

**Videos:**Sequences, Series and Power Series Videos

**Examples:**Sequences, Series and Power Series examples

**Maple:**

-:Convergence tests

-:Power series

-:Problem 9.3.28

-:Problem 9.5.12

**Exercises:** Sequences, Series and Power Series exercises

**Project problems:** Sequences, Series and Power Series project problems

**Maple TA test:** Maple TA