Paper foldingObserve what happens as I rip and generously share portions of my paper with you.

____ + _____ + ______ +______+_____+ ……. = ________

Is this an arithmetic series? Why or why not?

Geometric SeriesIn a geometric series the terms form a _________________________.

The terms of a geometric sequence for a ________________________.

What is the common ratio?Example: 4+ 12 + 36+……..

The common ratio of the terms is _________.

Unlike an arithmetic series, a geometric series can approach a certain number as more terms are added to it.

Converging vs. Diverging• Any geometric series that converges must have a common ratio less then

one.

• Any geometric series that diverges will have a common ratio greater then one.

To summarize:

Converges if: Diverges if:

|r| < 1Will approach a certain number.

|r| ≥ 1Will get infinitely big (or

infinitely small)

Note: A diverging series that has r ≥ 1 diverges, however if |r| is close to one but still greater then 1, it can diverge really slowly.

for example r=1.01 in the series below:→ 1+1.01+1.0201+1.030301+…….

grows slowly but since every new term grows the series by more then one, the more terms that get added on the bigger the number will become.

Do the following infinite geometric series diverge or converge?

1) 1+

Common ratio:____________________________

2)

Common ratio:_____________________________

3) 4+2+1+………?

Common ratio:_____________________________

Converge or diverge?

Sum of a Geometric SeriesTo find the sum of a finite geometric series you can use the formula:

Where is the first number in the sequence, r is the common ration and n is the number of terms.

Example: Use the formula to evaluate the following series to 8 terms: 5+15+45+135+……….,n=8

Common ratio:_____________

𝑆𝑛=𝑎1(1−𝑟𝑛)  1−𝑟

More Examples1) Calculate the number of triangles in a Sierpinski triangle

after stage 15.

2) Calculate the area of each additional triangle after stage 15