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

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

    • is the first term, 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

  • For example the sequence above can be written as

    • where

  • , , etc.

• A sequence may also be denoted using brackets

  • For example the sequence

    • where

• For this course all sequences are assumed to be infinite

  • I.e. goes from 1 to infinity

• Note that it is also possible to begin a sequence with

  • I.e. so that

    • is the first term, is the second term, etc.

    • This might make the formula for 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

  • 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, , associated with each series

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

  • I.e. , , , 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 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 , where is a real number, then we say that

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

• 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 and are both convergent series, and if and are constants, then the following results are true

  • 

  • 

  • 

    • 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