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If S denotes the sum to infinity and Sn ...

If `S` denotes the sum to infinity and `S_n` the sum of `n` terms of the series `1+1/2+1/4+1/8+ ,` such that `S-S_n<1/(1000),` then the least value of `n` is `8` b. `9` c. `10` d. 11

A

8

B

9

C

10

D

11

Text Solution

AI Generated Solution

To solve the problem, we need to find the least value of \( n \) such that the difference between the sum to infinity \( S \) and the sum of the first \( n \) terms \( S_n \) is less than \( \frac{1}{1000} \). ### Step-by-Step Solution: 1. **Identify the series**: The given series is \( 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots \). This is a geometric series with the first term \( a = 1 \) and common ratio \( r = \frac{1}{2} \). 2. **Calculate the sum to infinity \( S \)**: The formula for the sum to infinity of a geometric series is given by: \[ ...
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