LESSON X - Mathematics & Statistics
LESSON 19 CONVERGENT AND DIVERGENT SERIES
Definition Let [pic] be a sequence with a domain of [pic], where N is a positive integer. Then the sum of all the terms of in the sequence, denoted by [pic], is called an infinite series, or simply a series.
NOTE: If the domain of the sequence is [pic], then the series of this sequence is [pic].
Examples The following are series.
1. [pic] 2. [pic] 3. [pic]
4. [pic] 5. [pic] 6. [pic]
7. [pic] 8. [pic]
Definition Given the series [pic], define the sequence [pic], where [pic]. This sequence is called the sequence of partial sums. Note that the N given above is a positive integer.
Thus, given the series [pic], we have that
[pic]
[pic]
[pic]
[pic]
[pic]
.
.
.
[pic]
.
.
.
Thus, given the series [pic], we have that
[pic]
[pic]
[pic]
[pic]
[pic]
.
.
.
[pic]
.
.
.
Definition Given the series [pic], whose sequence of partial sums is [pic], then if [pic], then we say that the series is convergent (or converges, or converges to S.) If [pic] does not exist, then we say that the series is divergent (or diverges.) Note that the N given above is a positive integer.
TERMINOLOGY: S is called the sum of the series [pic] and write [pic]
For the examples given above, we will show in this lesson that the series [pic] and [pic] are divergent, and the series [pic] and [pic] are convergent. We will show in later lessons that the series [pic] is divergent, and the series [pic], [pic], and [pic] are convergent.
Definition The series [pic], where a and r are constants and [pic], is called a geometric series.
Theorem The geometric series [pic] converges and has a sum of [pic] if [pic]. The geometric series diverges if [pic].
Proof Will be provided later.
Example Determine whether the series [pic] converges or diverges. If it converges, then find its sum.
The series [pic], which was one of our examples given above, is a geometric series since [pic] = [pic].
Since [pic], then by the theorem above, this geometric series converges and has a sum of [pic] = [pic].
Answer: Converges; [pic]
Example Determine whether the series [pic] converges or diverges. If it converges, then find its sum.
This series was one of our examples given above.
We will rewrite the fraction [pic] using partial fraction decomposition.
[pic] = [pic] + [pic] [pic]
To solve for A, choose [pic]: [pic]
To solve for B, choose [pic]: [pic]
Thus, [pic] = [pic] = [pic] + [pic] = [pic].
Thus, [pic] = [pic] = [pic]
We will find the sequence [pic] of partial sums for the series [pic], where [pic]. Thus,
[pic]
[pic]
[pic]
[pic]
[pic]
.
.
.
[pic]
.
.
.
Then [pic] = [pic] = [pic] = [pic].
Thus, [pic]. Thus, [pic] = [pic] =
[pic] = [pic]
Answer: Converges; [pic]
COMMENT: The series [pic] is called a telescoping series.
Theorem If a series [pic] is convergent, then [pic].
Proof Will be provided later.
The contrapositive statement of this theorem gives us a test for divergence.
Test for Divergence: If [pic], then the series [pic] is divergent.
COMMENT: The Test for Divergence is the second most misused statement by Calculus students. Students want to apply the converse of the previous theorem, which is the statement if [pic], then the series [pic] is convergent. However, this is NOT true.
An easy example to keep in mind is the series [pic]. We have that [pic]. However, we will show in a later lesson that this series is DIVERGENT. The series [pic] is called the (divergent) harmonic series.
Examples Use the Divergence Test to show that the following series diverge.
1. [pic]
This series was one of our examples given above. We want to show that [pic].
[pic] = [pic]
Since [pic] = [pic], then we can write [pic] = [pic], which has an indeterminate form of [pic].
We will apply L’Hopital’s Rule to [pic]. Thus,
[pic] = [pic] = [pic] =
[pic] = [pic] = [pic]
Thus, [pic]. Thus, [pic].
Answer: Divergent (by the Divergence Test)
2. [pic]
This series was one of our examples given above.
[pic]
Answer: Divergent (by the Divergence Test)
Theorem If [pic] and [pic] are convergent series with sums A and B, respectively, then
1. [pic] is a convergent series and has of sum of [pic].
2. if c is a constant, then [pic] is a convergent series and has of sum of [pic].
3. [pic] is a convergent series and has of sum of [pic].
Proof Will be proved later.
Theorem If [pic] is a convergent series and [pic] is a divergent series, then [pic] is a divergent series.
Proof Will be provided later.
Examples Determine whether the following series converge or diverge. If the series converges, then give its sum.
1. [pic]
NOTE: This series can also be written as [pic].
This is a geometric series where [pic] and [pic]. Since [pic], then the geometric series converges and has a sum of
[pic] = [pic].
Answer: Converges; [pic]
2. [pic]
This is a geometric series where [pic] and [pic]. Since [pic], then the geometric series converges and has a sum of
[pic] = [pic].
Answer: Converges; [pic]
3. [pic]
[pic] = [pic] = [pic] = [pic]
This is a geometric series where [pic] and [pic]. Since [pic], then the geometric series diverges.
Answer: Diverges
4. [pic]
[pic] = [pic] = [pic] = [pic]
Thus, the series [pic] diverges by the Divergence Test.
Answer: Diverges
5. [pic]
We will rewrite the fraction [pic] using partial fraction decomposition.
[pic] = [pic] + [pic] [pic]
To solve for A, choose [pic]: [pic]
To solve for B, choose [pic]: [pic]
Thus, [pic] = [pic] = [pic].
Thus, [pic] = [pic]
We will find the sequence [pic] of partial sums for the series [pic]. Thus,
[pic]
[pic]
NOTE: [pic]
[pic]
NOTE: [pic]
.
.
.
[pic]
.
.
.
Then [pic] = [pic] = [pic] = [pic].
Thus, [pic]. Thus, [pic] =
[pic] = [pic]
Answer: Converges; [pic]
6. [pic]
We will rewrite the fraction [pic] using partial fraction decomposition.
[pic] = [pic] + [pic] [pic]
To solve for A, choose [pic]: [pic]
To solve for B, choose [pic]: [pic]
Thus, [pic] = [pic] = [pic] + [pic] =
[pic].
Thus, [pic] = [pic] = [pic]
We will find the sequence [pic] of partial sums for the series [pic]. Let [pic] Thus,
[pic]
NOTE: [pic]
[pic]
NOTE: [pic] = [pic]
[pic]
NOTE: [pic] = [pic]
[pic] = [pic]
NOTE: [pic] = [pic]
[pic] = [pic]
NOTE: [pic] = [pic]
[pic] = [pic]
NOTE: [pic] = [pic]
.
.
.
[pic] =
[pic] = [pic]
.
.
.
Then [pic] = [pic] = [pic].
Thus, [pic]. Thus, [pic] =
[pic] = [pic] = [pic]
Answer: Converges; [pic]
7. [pic]
We will rewrite the fraction [pic] using partial fraction decomposition.
[pic] = [pic] + [pic] [pic]
To solve for A, choose [pic]: [pic]
To solve for B, choose [pic]: [pic]
Thus, [pic] = [pic] = [pic] + [pic] =
[pic]. Thus,
[pic] = [pic] = [pic]
We will find the sequence [pic] of partial sums for the series [pic]. Let [pic] Thus,
[pic]
NOTE: [pic]
[pic]
NOTE: [pic] = [pic]
[pic]
NOTE: [pic] = [pic]
[pic] = [pic]
NOTE: [pic] = [pic]
[pic] = [pic]
NOTE: [pic] = [pic]
[pic] = [pic]
NOTE: [pic] = [pic]
.
.
.
[pic] =
[pic] = [pic]
.
.
.
Then [pic] = [pic] = [pic].
Thus, [pic]. Thus, [pic] =
[pic] = [pic] = [pic]
Answer: Converges; [pic]
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