The sum of 1 - 1 + 1 - 1 + 1 - 1 ... to even number of terms is —

The correct option is zero.

Understanding the Problem:
We are asked to find the sum of the series:

$1-1+1-1+1-1+\dots$

when taken to an even number of terms.

Step-by-Step Analysis:
Let us analyze the sum by grouping the terms in pairs:

• If we take 2 terms (an even number of terms):
$S_2=(1-1)=0$

• If we take 4 terms (an even number of terms):
$S_4=(1-1)+(1-1)=0+0=0$

• If we take 6 terms (an even number of terms):
$S_6=(1-1)+(1-1)+(1-1)=0+0+0=0$

In general, for any even number of terms $n=2k$ (where $k$ is a positive integer), the terms can be paired into $k$ pairs of $(1-1)$:

$S_{2k}=\underset{k\text{pairs}}{\underbrace{(1-1)+(1-1)+\dots +(1-1)}}=0$

Therefore, the sum of the series to an even number of terms is equal to zero.