Find the sum of the following series to infinity :
We observe that above progression possess a common ratio, but alternatively , adjacent terms are not possessing a common ratio. So, it consists of 2 geometric progressions.
Let, S =
⇒ S =
Let us denote the two progressions with S1 and S2
∴ S = S1 + S2
S1 =
Common ratio = r =
Sum of infinite GP = ,where a is the first term and r is the common ratio.
Note: We can only use the above formula if |r|<1
Clearly, a = and r = 1/9
⇒ S1 =
S2 =
Common ratio = r =
Sum of infinite GP = ,where a is the first term and r is the common ratio.
Note: We can only use the above formula if |r|<1
Clearly, a = and r = 1/25
⇒ S2 =
Hence,
S =