Question

How do you solve the series (7 - 9 + (81/7) - (729/49) + ... )?

7 - 9 + (81/7) - (729/49) + ...

How would you determine whether this is convergent or divergent? I'm not really sure where to even start. I think it's a geometric series? But I don't know how to get the formula for it...

Answer 1

Diverges.

Explanation:

Write as an actual series first.

So, the terms are alternating, with the first term being positive. `(-1)^(n-1)` must be present. It will definitely be an alternating series.

The third and fourth terms in the denominators are `7, 7^2=49,` respectively. So the denominator will have something to do with an exponentiated `7.`

The third and fourth terms in the denominator are `9^2=81, 9^3=729,` respectively. So, the numerator will have something to do with an exponentiated `9.`

We see pretty clear patterns for the third and fourth terms, but the first and second terms are outliers, there is no obvious pattern.

So, let's leave the first two terms alone and start our summation at `81/7.` To do this the summation will have to start at three, as `81/7` is the third term.

`7-9+sum_(n=3)^oo(-1)^(n-1)9^(n-1)/7^(n-1)`

`(-1)^(n-1)9^(n-1)=(-9)^(n-1)`. Apply this to the series:

`7-9+sum_(n=3)^oo(-9)^(n-1)/7^(n-1)`

Furthermore, `(-9)^(n-1)/7^(n-1)=(-9/7)^(n-1)`:

`7-9+sum_(n=3)^oo(-9/7)^(n-1)`

This is a geometric series in he form `sumr^n`, as we have a common ratio raised to a power. We could change the starting index to zero and rewrite it in the form `sum_(n=0)^ooa(r^n)` in an attempt to solve, but first, let's recognize that here, `r=-9/7`.

For a geometric series, if `|r|>1,` the series diverges. For our series, `|r|=|-9/7|=9/7>1,` so the series diverges. No further work is needed.

Answer 2

This series is divergent

Explanation:

Here, we notice that the terms share a common ratio of `-9/7`
Since the ratio is no within the range between `1` and `-1`, this series diverges.

If you try out the first few terms, you will find that the sum/difference keeps getting further and further away from each other, approaching nothing.

Answer 3

The series is divergent.

Explanation:

Since
`-9 = 7 times (-9/7)`
`81/7 = -9 times (-9/7)`
`-729/49 = 81/7 times (-9/7)`

This is a GP with the first term `a=7` and common ratio `r = -9/7`

Since `|r| = 9/7 >1` the infinite GP is divergent.