The sum of the following infinite series is \(2 + \frac{1}{2} + \frac{1}{3} +…
2024
The sum of the following infinite series is
\(2 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{8} + \frac{1}{9} + \frac{1}{16} + \frac{1}{27} + \ldots \)
- A.
11/3
- B.
7/2
- C.
13/4
- D.
9/2
Attempted by 3 students.
Show answer & explanation
Correct answer: B
Observation: the terms after the initial 2 are powers of 1/2 and powers of 1/3: 1/2, 1/3, 1/4, 1/8, 1/9, 1/16, 1/27, …
Group the series as the sum of three parts:
The initial term: 2
The powers of 1/2: 1/2 + 1/4 + 1/8 + … = 1
The powers of 1/3: 1/3 + 1/9 + 1/27 + … = 1/2
Compute each geometric sum quickly using a/(1−r): for 1/2-series a = 1/2, r = 1/2 gives 1; for 1/3-series a = 1/3, r = 1/3 gives 1/2.
Add the three parts:
Total = 2 + 1 + 1/2 = 7/2.