Find the sum of the infinite series 1/3+4/9+16/27+64/81+... if it exists

Question

campbell_st · Answer

well it would appear that the common ratio is 

$$r = \frac{4}{3}$$

since r > 1 then the series will diverge...

anonymous · Answer

what do u mean if it exists?

campbell_st · Answer

well if the series has a value of r such that 

-1 < r < 1 

then the series converges and you can find an infinite sum

anonymous · Answer

first find q.
$$q=\frac{ 4 }{ 9 }\div \frac{ 1 }{ 3 }=\frac{ 4 }{ 3 }$$
when q>1 we use this formula
$$s _{n}=\frac{ q^n-1}{ q-1 }$$
but we don't have n so we can't find the sum.

aum · Answer

This is an infinite series.  n -> infinity.

aum · Answer

Since this is a divergent series there is no sum.

anonymous · Answer

thank you

campbell_st · Answer

glad to help