Use the formula for the sum of a geometric series to find the sum or state that the series diverges.

(4/9)^n starting at n=-4

well, you recall from Algebra II that


∑ (4/9)^n = 1/(1 - 4/9) = 9/5
n=0

So just add on the first 4 terms from n = -4

Or, just start with a=(4/9)^-4 and use that in a/(1-r) for the sum

To find the sum of a geometric series, we can use the formula:

S = a / (1 - r),

where S represents the sum of the series, a is the first term, and r is the common ratio.

In this case, we have the series (4/9)^n, starting at n = -4.

To apply the formula, we need to determine the values of a and r.

The first term, a, can be found by substituting n = -4 into the given series:

a = (4/9)^(-4)

To simplify this, we can rewrite it as:

a = (9/4)^4

Next, we need to determine the common ratio, r. The common ratio is found by dividing any term of the series by the previous term.

In this case, we can calculate:

r = (4/9)^(-3) / (4/9)^(-4)

Simplifying this expression, we get:

r = (4/9)^(-3+4)
r = (4/9)^1
r = 4/9

Now that we have determined the values of a and r, we can substitute them into the formula to find the sum of the series:

S = a / (1 - r)
S = (9/4)^4 / (1 - 4/9)

Simplifying this further, we get:

S = (81/4) / (5/9)
S = (81/4) * (9/5)
S = 729/20

Therefore, the sum of the series (4/9)^n, starting at n = -4, is 729/20.