I need steps on how to complete this please i am so confused and lost. :(

Consider the infinite geometric series x e n=1 -4(1/3) n-1. In this image, the lower limit of the summation notation is "n=1".

a. Write the first four terms of the series.
b. Does the series diverge of converge?
c. If the series has a sum, find the sun.

If you mean the nth term is

-4 (1/3)^(n-1)

then clearly you have a GP where

a = -4
r = 1/3

The terms are

-4, -4/3, -4/9, ...

The sum is thus
a/(1-r) = -4/(2/3) = -6

I'm here to help you understand how to complete the given problem. Let's break it down step by step.

a. Write the first four terms of the series:
To find the first four terms of the series, we can substitute the values of n into the formula for the general term of the series. The general term of the given series is x * (-4/3)^(n-1), where n starts from 1.

So, let's calculate the first four terms using the given values:
- Term 1 (n = 1): x * (-4/3)^(1-1) = x * (-4/3)^0 = x * 1 = x

- Term 2 (n = 2): x * (-4/3)^(2-1) = x * (-4/3)^1 = -4x/3

- Term 3 (n = 3): x * (-4/3)^(3-1) = x * (-4/3)^2 = 16x/9

- Term 4 (n = 4): x * (-4/3)^(4-1) = x * (-4/3)^3 = -64x/27

Therefore, the first four terms of the series are: x, -4x/3, 16x/9, -64x/27.

b. Does the series diverge or converge?
To determine if the series converges or diverges, we need to check if the common ratio, which is (-4/3), is between -1 and 1.

In this case, the common ratio is -4/3, which is less than 1 (but not equal to -1). Therefore, the series converges.

c. If the series has a sum, find the sum:
To find the sum of the series, we can use the formula for the sum of an infinite geometric series. The formula is: S = a / (1 - r), where S is the sum, a is the first term, and r is the common ratio.

In this case, the first term a is x and the common ratio r is -4/3.

So, plugging in these values into the formula, the sum of the series is: S = x / (1 - (-4/3)) = x / (1 + 4/3)

Simplifying further, S = x / (3/3 + 4/3) = x / (7/3) = (3x) / 7.

Therefore, if the series has a sum, the sum would be (3x) / 7.

I hope this explanation helps you understand how to solve the problem. Let me know if you have any further questions!