Question

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

Answer 1

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"`

Answer 2

`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`