Question

Where do I even begin solving this problem?
∞
∑ -4 (1/3) n-1
n-1
a. Write the first four terms of the series.
b. Does the series diverge or converge?
c. If the series has a sum, find the sum.

Answer 1

First of all let me fix your expression to what I think it should say:

∞
∑ -4 (1/3)^(n-1)
n = 1

this expands to
-4(1/3)^0 + -4(1/3)^1 + -4(1/3)^2 + -4(1/3)^3 + .....
= -4 - 4/3 - 4/9 - 4/27 - ....

this is a geometric series with a = -4 and r = 1/3

it converges and has a sum of:

S_∞ = a/(1-r) = -4/(1-1/3)
= -4/(2/3)
= -6

notice the sum of the first 4 terms is already = -5.926