Approximate the sum of the series by using the first six terms.?

Question

 Approximate the sum of the series by using the first six terms.

Answer 1

 an = (-1)^(n + 1)n/3^n

 S6 

 = 1/3 - 2/9 + 1/9 - 4/81 + 5/243 - 2/243

 = 2/9 - 9/243

 = 45/243

 = 5/27

Answer 2

(-1)^(1 + 1) * 1 / 3^1 + (-1)^(1 + 2) * 2 / 3^2 + (-1)^(1 + 3) * 3 / 3^3 + (-1)^(1 + 4) * 4 / 3^4 + (-1)^(1 + 5) * 5 / 3^5 + (-1)^(1 + 6) * 6 / 3^6 =>

1/3 - 2/9 + 3/27 - 4/81 + 5/243 - 6/729 =>

(243 - 2 * 81 + 3 * 27 - 4 * 9 + 5 * 3 - 6) / 729 =>

(243 - 162 + 81 - 36 + 15 - 6) / 729 =>

(81 + 81 - 42 + 15) / 729 =>

(162 - 42 + 15) / 729 =>

(120 + 15) / 729 =>

135 / 729 =>

27 * 5 / (27 * 27) =>

5/27

Answer 3

sum_(n=1)^6 ((-1)^(n + 1) n)/(3 n) = 0