Sect 2.9 Infinite Geometric Series

SECT. 2.9INFINITE GEOMETRIC SERIES

I) WHAT IS AN INFINITE GEOMETRIC SERIES?

A Geometric series with an infinite number of terms

If the common ratio is greater than 1, each term will getbigger and the sum will soon add up to infinity

Note: Any infinite G.S. with a common ratio then the sum willadd up to infinity

Most infinite G.S. in sect. 2.9 will have a common ratiobetween -1 and 1

Each term in the series will get smaller

Eventually some terms will become so small such thatadding them will be insignificant, like adding zero

When the ratio is the infinite G.S. will converge to afixed value

These values are so small such that adding them will be insignificant!!

II) FORMULA FOR THE SUM OF AN INFINITE G.S.

If the Ratio is bigger than 1, the sum will be positive infinity

If the ratio is smaller than -1, the sum will be negativeinfinity

If the Ratio is between 1 and -1, the sum can be obtainedthrough a formula:

Formula for the sum of a G.S. with “n” terms

If there are infinite terms, then “n” will be infinity

Since

the value of

will be zero

The equation will only apply if

EX: FIND THE SUM OF THE INFINITE G.S.

EX: THE COMMON RATIO OF AN INFINITE GEOMETRICSERIES IS 0.75. IF THE SUM OF ALL THE TERMSCONVERGES TO 20, FIND THE FIRST TERM.

GIVEN THE FOLLOWING INFINITE GEOMETRICSEQUENCE, WHAT SHOULD THE VALUE OF “R”BE SO THAT THE SUM WILL BE 20?

The series must be converging

So the value of “r” must be between -1 and 1

The answer must be r = 0.80

EX: A MOVIE IN A THEATRE GENERATE $250,000 INREVENUE IN THE FIRST MONTH OF ITS SHOWING. EACHMONTH, SALES FROM THAT MOVIE DROP BY 15%. IF THEMOVIE IS SHOWN FOR A LONG TIME, WHAT IS THE TOTALREVENUE GENERATED?

HOMEWORK: