Page 28 - Math SL HB Sem 1

SIloUENCE AND SERIES

THE SUM'I'O INFINITY OFA GP

Consider the general GP, u.,, u,r, tl,r, ...
,
Now gn = z' (l-r") and
l-r

if lrf < r, then limr" :0, so l,g S, : :,"4? = #

if lrl > t, then limr" : oo and the series does not converge

Therefore, provided that < 1, a Gp converges to a sum of ]-
lrl

i.e fora CP,S-: a provided that . t.
l*r' lrl

Example I : Determine whether each series converges If it does, give its surr to
infinity
p1t*2*!n

Example 2: Find the condition sarisfied by x so that converges.
2.*
:
Evaluate this expression when x I .5

2-t6