Page 28 - Math SL HB Sem 1

StroTJI]NCE AND SERIES

THE SUM TO INFINITY OF A GP

Consider the general CP, u, urr, u,r, ...
,
Now ,n _ z, (l -r") and
l-r
:
-
if lrl < t, then tim r' 0, so l,:l S = lil (l r') =U,
"
l-r
l-r
if lrl > t, then limr'= "o and the series does not converge
Therefore, provided that < l, a Gp converges to a sum o1
lrl -4, .
l-r
ul
i.e for a GP,S = provided that < t
_ lrl
I -r

Example l: Determine whether each series converges. If it does, glve rts sul]'l to
infinity
o )'7
(u) t-1*l-l*.. 1+:
41664 Q)) 3+ +

Example 2: Find the condition sarisfied by x so that
Z+ converges
rO
:
Evaluate this expression when x 1.5

2-t6