Question

The sum to \( n \) terms of the series
\( \dfrac{1}{2} + \dfrac{3}{4} + \dfrac{7}{8} + \dfrac{15}{16} + \cdots \)
is

• A. \(n-1-2^{-n}\)
• B. \(1\)
• C. \(n-1+2^{-n}\)
• D. 1+2⁻n

Hint:

Write terms as 1-(geometric term) to simplify sums.

Answer 1

The Correct Option is C

Step 1: General term:
\( T_k = \dfrac{2^k - 1}{2^k} = 1 - \dfrac{1}{2^k} \)
Step 2: Sum:
\( S_n = \sum_{k=1}^{n} \left(1 - \dfrac{1}{2^k}\right) \)
\( = n - \sum_{k=1}^{n} \dfrac{1}{2^k} \)
Step 3:
\( \sum_{k=1}^{n} \dfrac{1}{2^k} = 1 - \dfrac{1}{2^n} \)
\( \Rightarrow S_n = n - 1 + \dfrac{1}{2^n} \)