Find the sum of the series. Enter the exact answer.

Question

- 4 + 1 - 1/4 + 1/16 - 1/64 + 1/256 - ... =

Answer 1

The series - 4 + 1 - 1/4 + 1/16 - 1/64 + 1/256 - ...

Above series is geometric series.

Common ratio r = t₂/t₁

r = 1/(- 4)

r = - 1/4

In this case r < 1.

Formula for the sum of the sequence S_n = a(1 - rⁿ)/(1 - r)

Where r is common ratio. a is first term.

Substitute r = (- 1/4) and a = - 4 in S_n .

S_n = [(- 4)(1 - (- 1/4)ⁿ]/[1 - (- 1/4)]

S_n = [(- 4)(1 - (- 0.25)ⁿ]/[1 - (- 0.25)]

S_n = [(- 4)(1 - (- 0.25)ⁿ]/[1 + 0.25]

S_n = (- 4 + 4 (- 0.25)ⁿ)/1.25

    = 3.2 (- 1 + (- 0.25)ⁿ)