Question

The series \( 1 + 1 + \frac{3}{2^2} + \frac{4}{2^3} + \frac{5}{2^4} + \cdots \) is equal to

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

In series problems, breaking down the terms and using known summation formulas helps simplify the problem.

Answer 1

The Correct Option is D

This is a series of the form: \[ S = 1 + 1 + \frac{3}{2^2} + \frac{4}{2^3} + \frac{5}{2^4} + \cdots \] Recognizing the pattern, we can express this as: \[ S = 1 + 1 + \sum_{n=2}^{\infty} \frac{n+1}{2^n} \] Using the sum of the geometric series and the properties of the terms, we calculate the total sum: \[ S = 4 \]