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Wize University Calculus 2 Textbook > Sequences and Series

Basic Definitions

Comparison Theorem for Improper IntegralsPrevious Section
Test of DivergenceNext Section
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Sequences

A sequence is just an ordered collection of numbers, a1, a2, a3, ...a_1,\ a_2,\ a_3,\ ...

Common Sequences
  • Arithmetic Sequence
  • Starts with aa, then adds a constant dd to get the next term
  • Example: 1, 4, 7, 10, 13,...
  • The nth term is a+(n−1)da+(n-1)d
  • Geometric Sequence
  • Starts with aa, then multiply by a ratio rr to get the next term
  • Example: 2, 4, 8, 16, 32,...
  • The nth term is arn−1ar^{n-1}
  • Harmonic Sequence
  • 1, 12, 13, 14, ...1,\ \frac{1}{2},\ \frac{1}{3},\ \frac{1}{4},\ ...


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Sequence Convergence

A sequence {a1, a2, a3,...,an,...}\left\{a_1,\ a_2,\ a_3,...,a_n,...\right\} is said to converge if the terms eventually approach a finite number. In other words, the sequence converges if lim⁡n→∞an=L{\displaystyle \lim_{n\to\infty}}a_n=L, where LL is a finite number.

Otherwise, we say that the sequence diverges.

Examples
  • The harmonic sequence an=1na_n=\frac{1}{n} converges to 0 because lim⁡n→∞an=0{\displaystyle \lim_{n\to\infty}}a_n=0
  • The sequence {cos⁡π, cos⁡2π, cos⁡3π, cos⁡4π,...}={−1, 1, −1, 1,...}\left\{\cos\pi,\ \cos2\pi,\ \cos3\pi,\ \cos4\pi,...\right\}=\left\{-1,\ 1,\ -1,\ 1,...\right\} diverges because the terms oscillate back and forth between -1 and 1, it never approaches a certain value (i.e. the limit does not exist!)

Definitions
  • If the terms inside a sequence are always increasing (each term gets bigger and bigger) OR always decreasing (each term gets smaller and smaller), we call this sequence a monotonic sequence.
  • If a sequence is bounded above by a certain finite number (all the terms are smaller than this number) AND if the a sequence is bounded below by a certain finite number (all the terms are bigger than this number), we call this sequence a bounded sequence.


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Note-Worthy Theorems

1. Squeeze Theorem for Sequences:

Say we have 3 sequences:{an},  {bn},  {cn}\left\{a_n\right\},\ \ \left\{b_n\right\},\ \ \left\{c_n\right\}.
Suppose that after a certain points, all the terms in one sequence has values that are sandwiched between the terms of the other two sequences (i.e. an≤bn≤cna_n\le b_n\le c_n). Then if lim⁡n→∞an=lim⁡n→∞cn=L{\displaystyle \lim_{n\to\infty}}a_n={\displaystyle \lim_{n\to\infty}}c_n=L, then lim⁡n→∞bn=L{\displaystyle \lim_{n\to\infty}}b_n=L. In other words, if the sandwiching sequences converge to the same finite number, the middle sandwiched sequence must also converge to that value.

2. Monotonic Sequence Theorem

Every bounded, monotonic sequence is convergent
In plain English:
1.) if you can find a number that is bigger than all the terms in your sequence,
2.) if you can find a number that is smaller than all the terms in your sequence, and
3.) the terms inside the sequence are either always getting larger or always getting smaller,
then the sequence will converge to a finite number.

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Series

For a given sequence a1, a2, a3, ...a_1,\ a_2,\ a_3,\ ..., we define the series to be the formal sum ∑n=1∞an{\displaystyle \sum^\infty_{n=1}}a_n
  • The Nth-partial sum of the series is SN=∑n=1NanS_N={\displaystyle \sum^N_{n=1}}a_n
  • A series ∑n=1∞an{\displaystyle \sum^\infty_{n=1}}a_n converges if lim⁡N→∞SN= lim⁡N→∞∑n=1Nan=L{\displaystyle \lim_{N\to\infty}}S_N=\space {\displaystyle \lim_{N\to\infty}}{\displaystyle \sum^N_{n=1}}a_n=L (i.e. a finite number)
  • The series diverges if this limit does not exist or approaches infinity.


Note: If ∑n=1∞an{\displaystyle \sum^\infty_{n=1}}a_n converges, then so does −k∑n=1∞an-k{\displaystyle \sum^\infty_{n=1}}a_n.



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Common/Known Series (memorize!)
Geometric Series
∑n=1∞arn−1{\displaystyle \sum^\infty_{n=1}}ar^{n-1} or ∑n=0∞arn{\displaystyle \sum^\infty_{n=0}}ar^n
  • If ∣r∣<1|r|<1: converges to a1−r\frac{a}{1-r}
  • If ∣r∣≥1\left|r\right|\ge1: diverges
P-Series
∑n=1∞1np{\displaystyle \sum^\infty_{n=1}}\frac{1}{n^p}
  • If p>1p>1: converges
  • If p≤1p\le1: diverges
Harmonic Series
∑n=1∞1n{\displaystyle \sum_{n=1}^\infty}\frac{1}{n}
  • Diverges always (special case of p-series)

Example
Determine whether the following series converge or diverge:

a.) ∑n=1∞2(0.3)n−1\sum\limits^\infty_{n=1}2(0.3)^{n-1}
This is a geometric series with a=2, r=0.3a=2,\ r=0.3
Since ∣r∣<1\left|r\right|<1, the series converges.

b.)∑n=1∞1n3\sum\limits^\infty_{n=1}\frac{1}{n^3}
This is a p-series with p>1p>1. Therefore, the series converges.

Practice: Partial Sums

Suppose that the partial sums of the series ∑n=1∞an{\displaystyle \sum_{n=1}^\infty}a_n is given by SN=a1+a2+...+aN=2N−1N+1S_N=a_1+a_2+...+a_N=2N-\frac{1}{N+1}.
Determine whether the series converges or diverges.

Practice: Geometric Series

Determine whether the series ∑n=0∞(−1)n23n9n\sum\limits^\infty_{n=0}\frac{(-1)^n2^{3n}}{9^n} converges or diverges.
If it converges, determine the value of the sum.

Practice: Geometric Series

Suppose that ∑n=0∞rn{\displaystyle \sum_{n=0}^\infty}r^n converges to 5. Does ∑n=0∞(r−1)n{\displaystyle \sum_{n=0}^\infty}(r-1)^n converge or diverge?
If it converges, what does it converge to?

Practice Question

Write 1.123‾1.\overline{123} as a fraction pq\frac{p}{q}.

How to Test if a Series Converges?

In the next few sections, you will learn about the following tests that are used to determine if a series converges or diverges:
  • Test of divergence
  • Integral test
  • (Limit/Series) Comparison Test
  • Absolute Convergence Test
  • Alternating Series Test
  • Ratio Test
  • Root Test
In many cases, even if we determine that a series converges, we may not know what it converges to.