This week in Pre-calculus 11 we reviewed the first four units in preparation for the midterm. Though it was review, one thing that I had to go back and take a look at was lesson 1.5 [Infinite Geometric Series].

what to know:

• There are two types of geometric series, diverging [no sum] / converging [sum].

Diverging (no sum):

• when a geometric series diverges that means that the numbers are getting bigger
• If a geometric series diverges that means:
•  r>1
• r<-1
• If a geometric series diverges then it also means that it has no sum

Converging (sum):

• When a geometric series converges that means that the numbers are getting closer together
• If a geometric series converges that means:
• 0<r<1
• -1<r<0
• If a geometric series converges then that means that you can find its infinite sum:

Example:

• First, we need to find the common ratio to determine if this geometric series converges or diverges. 
• To find the common ratio, you take one of the terms and divide it by the preceding term.
• Once you find the common ratio compare it to the restrictions of both a converging and diverging geometric series. The common ratio in this equation is r = 4, 4>1 and therefore diverges. Since this geometric series diverges we can not find its infinite sum.

Example:

• First, we need to find the common ratio to determine if this geometric series converges or diverges. 
• To find the common ratio, you take one of the terms and divide it by the preceding term.
• Once you find the common ratio compare it to the restrictions of both a converging and diverging geometric series. The common ratio in this equation is r= 0.25 or . since 0<<1, this geometric series converges and therefore has an infinite sum.
• to calculate the infinite sum of this geometric series we use the formula:   and fill it is with what we know. Note: a is used to represent .
• Once you have solved the equation you should be left with the geometric series infinite sum.