Geometric series Sum to Infinity Geometric series Sum

Geometric series – Sum to Infinity

Geometric series – Sum to Infinity The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

Geometric series – Sum to Infinity We will find a formula for the sum of an infinite number of terms of a G. P. This is called “the sum to infinity”, e. g. For the G. P. we know that the sum of n terms is given by As n varies, the only part that changes is This term gets smaller as n gets larger. As n approaches infinity, We write: approaches zero. .

Geometric series – Sum to Infinity So, for For the series However, if, for example r = 2, As n increases, also increases. In fact, The geometric series with There is no sum to infinity diverges

Geometric series – Sum to Infinity Convergence If r is any value greater than 1, the series diverges. Also, if r < -1, ( e. g. r = -2 ), So, again the series diverges. If r = 1, all the terms are the same. If r = -1, the terms have the same magnitude but they alternate in sign. e. g. 2, -2, . . . A Geometric Series converges only if the common ratio r lies between -1 and 1. for ( or )

Geometric series – Sum to Infinity SUMMARY Ø A geometric sequence or geometric progression (G. P. ) is of the form Ø The nth term of an G. P. is Ø The sum of n terms is or Ø The sum to infinity is or