Decide if Converging or Diverging

Converge, Diverge, Oh My!

How can you decipher when a series will converge or diverge?

Follow the flow chart on the next page!

How do you know if a series is infinite?

Never ends is another word for infinite. So if it stops, it is finite, and if it never ends, it is infinite.

ALL finite series converge. Finite means there is an end.

Example 1 of a Finite Series.

23

∑3

n

n=0 Notice there is a start and end for the index n. This means each number from 0 through 23 needs substituted into the series.

Example 2 of a Finite Series.

2 + 4 + 6 + ...+ 48 The … mean there are more terms but it was just way too many to write!

Key is after the … there is a + 48 to show it stops at 48. Stop means it’s finite. Thus, it converges.

Checking if Arithmetic Series.

ALL infinite Arithmetic Series diverge but how can you tell? If each term changes by adding the same number, it is an Arithmetic Series.

Remember that adding a negative is the same as subtracting!

Is this Arithmetic or Geometric? ∞ n 3 ∑ n=0

Can’t tell by looking? Find a couple terms and then ‘test’ to determine if Arithmetic.

∞

∑3

n

n=0

Evaluate terms: 0

3 =1 31 = 3 2

3 =9 Find Common Difference: Term 2 - Term 1 =3-1 =2

Add each term by the Common Difference to verify you get the next term.

Term 2 = Term 1 + Common Difference =1+2=3 Term 2 was 3 so this checks out.

Term 3 = Term 2 + Common Difference =3+2=5 Term 3 was 9. Therefore, this does not check out.

Because it does not check out, it is not an Arithmetic Series. 

Check if a Geometric Series. ∞

∑3

n

n=0

Use the same approach as for Arithmetic if you cannot ‘see’ if this Geometric.

Evaluate terms:

30 = 1 31 = 3 32 = 9 Find Common Ratio: Term 2 / Term 1 = 3 / 1 = 3

Multiply each term by the Common Ratio to verify you get the next term.

Term 2 = Term 1 * Common Ratio =1*3=3 Term 2 was 3 so this checks out.

Term 3 = Term 2 * Common Ratio =3*3=9 Term 3 was 9 so this checks out.

Because each term is found by multiplying the previous term by the same number, this is a Geometric Series.