Find the sum of the infinite series if it exists. If it exists, round to one decimal place. If it does not exist, write none. 1/3+4/9+16/27+64/81

Question

Find the sum of the infinite series if it exists.  If it exists, round to one decimal place.  If it does not exist, write none. 1/3+4/9+16/27+64/81

tkhunny · Answer

$\dfrac{1}{3} + \dfrac{1}{3}\cdot\dfrac{4}{3} + \dfrac{1}{3}\cdot\left(\dfrac{4}{3}\right)^{2} + \dfrac{1}{3}\cdot\left(\dfrac{4}{3}\right)^{3} + ...$

Hmmm...  I'd worry about convergence before I tried to fidn the sum.

anonymous · Answer

wait so none exists or what??

tkhunny · Answer

What do you think?  A geometric series with r = 4/3 > 1.

anonymous · Answer

i dont think so but im not sure

tkhunny · Answer

|r|, the common ratio, must be less than 1 for convergence.

anonymous · Answer

ok?? im so confused

tkhunny · Answer

Confused about what?  It doesn't converge.  State that this is so and move on to the next problem.  This one is done.