Finding the sum of a series

Question

Given the series, ∑_n=0^∞ (1/4)(-2/5)^n

Show that the series converges

and compute the value of the sum of the series

Answer 1

it's a geometric series with common ratio, r, = -2/5

factor out the 1/4 of each term

(1/4)(1 - 2/5 + 4/25 - 8/125 + 16/625 -32/3125 + 64/15625 ....)

= (1/4)(1-.4 + .16 - .064 + .0256 - ,01024 + .00496 - ...)

= (1/4)(1.19056 - .47424)

= (1/4)(.71632) = .178... +, round off to .179

5/28 = about .179

sum of an infinite geometric series = first term divided by 1-r= a1/(1-r) = (1/4)/(1--2/5) = (1/4)/7/5) = 5/28 = about .179.

if -1< r < 1, then the geometric series converges

Answer 2

This series is a geometric series of the form ∑ g₁rⁿ, where r = -2/5. Using the geometric test, |r| < 1 so the series converges

g₁=1/4

S_n = g₁/(1-(r)) = (1/4)/(1- -2/5) = (1/4)/(1+2/5) = (1/4)/(7/5) = (1/4)(5/7) = 5/28