GATE 2024 CY – Question 4
The sum of the following infinite series is $$2+\frac12+\frac13+\frac14+\frac18+\frac19+\frac1{16}+\frac1{27}+\cdots$$
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Correct answer: (B) 7/2
Explanation
Group the terms into two interlaced geometric series.
$$2+\frac12+\frac13+\frac14+\frac18+\frac19+\frac1{16}+\frac1{27}+\cdots$$
- **Powers of 2:** $\dfrac12,\ \dfrac14,\ \dfrac18,\ \dfrac1{16},\dots$ plus the first term 1 (taken from the 2). The series $1+\dfrac12+\dfrac14+\cdots$ has sum $\dfrac{1}{1-1/2}=2$.
- **Powers of 3:** $1,\ \dfrac13,\ \dfrac19,\ \dfrac1{27},\dots$ with sum $\dfrac{1}{1-1/3}=\dfrac32$.
The leading 2 in the question is $1+1$, the first terms of both series.
$$\text{Sum}=2+\frac32=\frac72\quad(\text{option B}).$$