Let An=34−(34)2+(34)3−....+(−1)n−1(34)n
and Bn=1−An, then find the least value of n0, n0∈N such that Bn>An,∀n≥n0.
Given:
An=34−(34)2+(34)3−....+(−1)n−1(34)n
Clearly this series in G.P.
Here, a=34
r=(34)2(34)=34<1
An= sum of G.P. of n terms whose common radio is −34
An=a(1−rn)1−r
∴An=34[1−(−34)n]1−(−34)
=34.47.[1−(−34)n]
=37[1−(−34)n]
Bn=1−An and
Bn>An⇒1−An>An
⇒1>2An
⇒An<12
∴37[1−(−34)n]<12
⇒1−(−34)n<76
⇒1−76<(−34)n
⇒−16<(−34)n
⇒(−34)n>−16
Which is possible only for all n≥6.