Convergent & Divergent Infinite Series (College Board AP® Calculus BC): Revision Note

Sequences & series

What is a sequence?

• A sequence is an ordered collection of numbers

  • 'Ordered' means that there is a first number, second number, third number, etc.

  • Each number in a sequence is referred to as a term

  • For example, the sequence $\frac{1}{2},\frac{1}{4},\frac{1}{8},\frac{1}{16},...$

    • $\frac{1}{2}$ is the first term, $\frac{1}{4}$ is the second term, etc.

• A sequence is also a function whose domain is the positive natural numbers (1, 2, 3, 4, ...)

  • It may (but will not always) be possible to write the function explicitly as an expression in the term number $n$

  • For example, the sequence above can be written as $a_n=a\left(n\right)=\frac{1}{2^n}$

    • where $n=1,2,3,4,...$

  • $a_1=\frac{1}{2}$, $a_2=\frac{1}{4}$, etc.

• A sequence may also be denoted using brackets

  • For example, the sequence $\left\{a_n\right\}$

    • where $\left\{a_n\right\}=\left\{a_1,a_2,a_3,...,a_n,...\right\}$

• For this course all sequences are assumed to be infinite

  • I.e. $n$ goes from 1 to infinity

• Note that it is also possible to begin a sequence with $n=0$

  • I.e. so that $\left\{a_n\right\}=\left\{a_0,a_1,a_2,...,a_n,...\right\}$

    • $a_0$ is the first term, $a_1$ is the second term, etc.

    • This might make the formula for $a_n$ easier to write

  • If this is the case, it will be made clear in the question

What is a series?

• A series is the sum of all the terms in a sequence

  • For example the sum is the series associated with the sequence $\left\{a_n\right\}$

  • Note that this is the sum of an infinite number of terms

Convergent & divergent infinite series

What is a sequence of partial sums?

• There is a sequence of partial sums, $\left\{s_n\right\}=\left\{s_1,s_2,s_3,...\right\}$, associated with each series

  • $s_n$ in each case is the sum of the first n terms of the sequence that underlies the series

    • $s_1=a_1$

    • $s_2=a_1+a_2$

    • $s_3=a_1+a_2+a_3$, etc.

What does it mean for a series to converge or diverge?

• The series converges if its associated sequence of partial sums converges

  • I.e. if the limit $\underset{n\to \infty }{\lim }{s}_n$ exists and is non-infinite

  • In this case the series sum gets closer and closer to a fixed value as more and more terms are added

• If $\underset{n\to \infty }{\lim }{s}_n=S$, where $S$ is a real number, then we say that $\sum _{n=1}^{\infty }{a}_n=S$

  • I.e. the sum of the infinite series is equal to $S$

• If a series does not converge, then it diverges

  • I.e. the sum never settles down to a single fixed finite value, no matter how many terms are added on

Algebra of convergent series

How can I combine results for convergent infinite series?

• If $\sum _{n=1}^{\infty }{a}_n$ and $\sum _{n=1}^{\infty }{b}_n$ are both convergent series, and if $c$ and $d$ are constants, then the following results are true

  • $\sum _{n=1}^{\infty }c{a}_n=c\sum _{n=1}^{\infty }{a}_n$

  • $\sum _{n=1}^{\infty }\left(a_n\pm b_n\right)=\sum _{n=1}^{\infty }{a}_n\pm \sum _{n=1}^{\infty }{b}_n$

  • $\sum _{n=1}^{\infty }\left(c{a}_n\pm d{b}_n\right)=c\sum _{n=1}^{\infty }{a}_n\pm d\sum _{n=1}^{\infty }{b}_n$

    • The third result is just the combination of the first two

    • It can be extended to sums or differences of more than two series

• These results are not valid for divergent series