Question

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

Answer 1

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

Answer 2

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

Answer 3

Now do you think this series exists?

Answer 4

yes?

Answer 5

Well, this is geometric series, right?

Answer 6

And we have common ratio greater than 1.

Answer 7

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

Answer 8

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

Answer 9

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

Answer 10

Do you know calculus?

Answer 11

no

Answer 12

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?

Answer 13

yes

Answer 14

well, so when you "get" to infinity.

You would have infinity in numerator.

Answer 15

So this series doesn't exist.

Answer 16

Does that make sense?

Answer 17

Okay thank you <3