Series: determine convergence or divergence, and justify

[stars584]

Determine if following series converge or diverge. Justify your answer.
1.
Infin
Σ 12(1.6)^n
n=1

12(1.6)^1= 19.2 12(1.6)^2= 30.72 12(1.6)^3= 49.152
Therefore it diverges.
2.
Infin
Σ 6(.8)^n
n=1

6(.8)^1 = 4.8 6(.8)^2 = 3.84 6(.8)^3 = 3.072 6(.8)^4 = 2.4576

Therefore it diverges
3.
Infi
Σ n^2/(3n^2 + 1)
n=1

(1^2)/(3(1)^2 + 1) =1/4 (2^2)/(3(2)^2 + 1) = 4/13

3^2/(3(3)^2 + 1) = 9/28 4^2/(3(4)^2 + 1) = 16/49

Therefore it converges
2.
A point starting at the origin moves 1 unit to the right, 1/3 up, 1/9 left, 1/27 down, 1/81 to the right, 1/243 up , etc. In other words, after each move, the point makes a 90 degree left turn and moves 1/3 the distance of its previous move.
To what coordinates does the point converge?
(1 , 1 )

 

[skeeter]

Re: Series

Determine if following series converge or diverge. Justify your answer.

1.
Infin
Σ 12(1.6)^n
n=1

geometric series with r > 1 ... series diverges

2.
Infin
Σ 6(.8)^n
n=1

geometric series with 0 < r < 1 ... series converges.

3.
Infi
Σ n^2/(3n^2 + 1)
n=1

nth term does not go to 0 as n approaches infinity (it approaches 1/3) ... series diverges

2.
A point starting at the origin moves 1 unit to the right, 1/3 up, 1/9 left, 1/27 down, 1/81 to the right, 1/243 up , etc. In other words, after each move, the point makes a 90 degree left turn and moves 1/3 the distance of its previous move.
To what coordinates does the point converge?
(1 , 1 )

in the x-direction ...
1 - 1/9 + 1/81 - 1/729 + ... + (-1/9)^n-1 + ...

in the y-direction ...

1/3 - 1/27 + 1/243 - 1/2187 + ... =
(1/3)[1 - 1/9 + 1/81 - 1/729 + ... + (-1/9)^n-1 + ...]

find the sum of each geometric series to find the coordinates.