What are some examples of infinite geometric series?

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Answer 1 · 2015-07-13T18:38:56

Here are some examples:

#1 + 1/2 + 1/4 + 1/8 + 1/16 +...#

#1 - 1 + 1 - 1 + 1 - 1 +...#

#1 + 2 + 4 + 8 + 16 +...#

All geometric series are of the form #sum_(i=0)^oo ar^i# where #a# is the initial term of the series and #r# the ratio between consecutive terms.

In the three examples above, we have:

#a = 1#, #r = 1/2#

#sum_(i=0)^oo ar^i = 2#

#a = 1#, #r = -1#

#sum_(i=0)^oo ar^i# does not converge - it alternates between #0# and #1# as each term is added.

#a = 1#, #r = 2#

#sum_(i=0)^n ar^i -> oo# as #n->oo#

The geometric series #sum_(i=0)^oo ar^i# only converges in the following cases:

(1) #a = 0#

#sum_(i=0)^oo ar^i = 0#

(2) #abs(r) < 1#

#sum_(i=0)^oo ar^i = a/(1-r)#