I need help evaluating the geometric infinite series.

It's got the double zero thing over the E thing.

00
E(-1/2)^ n-1
(n=1)

Substitute:

n = k + 1

Then k runs from zero to infinity

The summand is (-1/2)^k

So, the summation is:

1/[1- (-1/2)] = 2/3

How did you get k + 1?

How would I do
00
E 3(0.4)^n-1
n=1

Would that be the same basic thing? Like 1/[1-3(0.4)]??

You are free to shift your summation variable in any way you like. If you put n = k + 1 then what happens is that the lower limit of the summation of

n = 1 corresponds to k = 0 while the upper limit remains infinity.

In the summand you then replace n by
k + 1 and you then see that the summation is now in the standard form of a sum from zero to infinity of a^k which is 1/(1-a)

To evaluate the given geometric infinite series, let's break down the expression and then use the formula for the infinite geometric series.

The given series is:
∑ (n=1 to ∞) 00 / (e(-1/2)^(n-1))

Here, the numerator is 00, and the denominator is (e(-1/2))^(n-1).

To evaluate the infinite geometric series, we can use the formula:

S = a / (1 - r)

where:
S is the sum of the series,
a is the first term of the series, and
r is the common ratio between consecutive terms.

In our case, the first term (a) is 00, which means it is a constant value of zero.

The common ratio (r) can be found by dividing any term by its previous term. Looking closely, each term is divided by e(-1/2) raised to the (n-1) power. So, to find the common ratio, we need to divide any term by its previous term.

Let's consider the second term (n = 2):
Term 2 = 00 / e(-1/2)^(2-1)
Term 1 = 00 / e(-1/2)^(1-1) = 00 / e^0 = 00 / 1 = 00

Now, we calculate the common ratio (r) by dividing Term 2 by Term 1:
r = Term 2 / Term 1 = (00 / e(-1/2))^1 / 00 = (00 / e(-1/2))

Since the common ratio (r) is the same for all terms, we can substitute these values into the formula for the infinite geometric series:

S = a / (1 - r)
S = 00 / (1 - (00 / e(-1/2)))

Note that anything divided by zero (00 / e(-1/2)) is undefined, so this series cannot be evaluated using the formula. It seems that there is an issue with the given series. Please check the expression and try to provide the correct terms and values.