Question

The sum of infinity of the series 2/3 – 4/3² + 2/3³ + 4/3⁴ is​

Answer 1

`(3)/(4) - (2)/(4) = (1)/(4)`

Answer 2

• 7/20

Explanation:

It seems there are two patterns in the series:

• 2/3, 2/3³, ... with a₁ = 2/3 and r = 1/9
• - 4/3², 4/3⁴, ... with a₁ = -4/9 and r = -1/9

Using sum of infinity of the geometric series formula:

• S∞ = a₁/(1 - r)

Applying this to both patterns:

• S₁ = 2/3÷(1 - 1/9) = 2/3 ÷ 8/9 = 2/3×9/8 = 3/4
• S₂ = -4/9÷(1 - (-1/9)) = -4/9÷10/9 = -4/9×9/10 = - 2/5

Sum of the two results:

• 3/4 + (-2/5) =
• 5/5*3/4 - 4/4*2/5 =
• 15/20 - 8/20 =
• 7/20