How do I find the sum of the infinite geometric series 1/2 + 1 + 2 + 4 +... ?

2 Answers
Jun 5, 2018

see below

Explanation:

for a GP the sum to infinity

S_(oo)=a/(1-r)

a=" first term, "r=" the common ratio "

only exists if

|r|<1

for the given GP

1/2+ 1+2+4+..

r=1/(1/2)=2

:.|r|>1

=> " sum to infinity does not exist"

Jun 9, 2018

S_n=1/2Sigma_(i=0)^n(2^i)->oo as n->oo, i.e. the sum is indefinite.

Explanation:

You can write the sum of n first terms in the geometric series as S_n=1/2Sigma_(i=0)^n(2^i)
It's a well known fact that 2^n->oo as n->oo. Therefore, as the individual terms go towards infinity, so must the sum.

Therefore S_n=1/2Sigma_(i=0)^n(2^i)->oo as n->oo