## anonymous one year ago Help please?? Find the sum of the infinite series 1/3 +4/9 + 16/27+ 64/81+...if it exists

1. geerky42

Looks like $\sum\dfrac{4^n}{3^{n+1}}$

2. geerky42

$\cdots=\dfrac{1}{3}\sum\dfrac{4^n}{3^n} =\dfrac{1}{3}\sum\left(\dfrac{4}{3}\right)^n$

3. geerky42

Now do you think this series exists?

4. anonymous

yes?

5. geerky42

Well, this is geometric series, right?

6. geerky42

And we have common ratio greater than 1.

7. anonymous

yeah but I need to know what the sum of the series is

8. geerky42

For any finite geometric series, formula is $\sum_{i=1}^{n-1} r^i = \dfrac{1-r^n}{1-r}$Right?

9. anonymous

I think so, but what do put in to get the answer? Can you help walk me through the steps? I'm really confused.

10. geerky42

Do you know calculus?

11. anonymous

no

12. geerky42

Really? ok well just take a look at $\dfrac{1-r^n}{1-r}$ Since common ratio (r) is greater than 1, what would happen to it if n getting bigger and bigger? $$r^n$$ would get bigger and bigger, right?

13. anonymous

yes

14. geerky42

well, so when you "get" to infinity. You would have infinity in numerator.

15. geerky42

So this series doesn't exist.

16. geerky42

Does that make sense?

17. anonymous

Okay thank you <3