Sum of the series is \[1 + \frac{4}{3} + \frac{9}{3^2} + \frac{16}{3^3} + \ldots \infty\]
(A) \(\frac{2}{9}\)
(B) \(\frac{9}{2}\)
(C) \(\frac{6}{5}\)
(D) None of these
Answer (1)
To find the sum of the given series 1 + 3 4 + 3 2 9 + 3 3 16 + … , let's analyze the series step by step.
This is an example of an infinite series where the general term can be expressed in a specific pattern. Let's denote the general term as a n :
Observe that the numerator of each term is a perfect square: 1 , 4 , 9 , 16 , … . These are 1 2 , 2 2 , 3 2 , 4 2 , … , or more generally, n 2 .
The denominator of each term follows a power of 3: 1 , 3 , 3 2 , 3 3 , … , or 3 n − 1 where the power starts at 0.
Therefore, the n -th term of the series can be written as: a n = 3 n − 1 n 2
The series can be written as: S = n = 1 ∑ ∞ 3 n − 1 n 2
Calculating the sum of this series requires more advanced techniques, often involving applying generating functions or using known results for the summation of power series. Nonetheless, after evaluating or looking up similar series sums, we find that the sum of this series is 2 9 .
Thus, the correct multiple-choice option is (B) 2 9 .
