Solving the Alternating Series Calculation

The problem requires calculating the sum of the series: $1 - 1 + 1 - 1 + 1 - 1 + \dots$ evaluated for 101 terms.

Step-by-Step Series Solution

We can solve this by observing the pattern or by pairing terms.

Method 1: Pattern Recognition

• 1 term: $1 = 1$
• 2 terms: $1 - 1 = 0$
• 3 terms: $1 - 1 + 1 = 1$
• 4 terms: $1 - 1 + 1 - 1 = 0$

The pattern shows that if the number of terms is odd, the sum is 1. If the number of terms is even, the sum is 0.

Since there are 101 terms (an odd number), the sum is 1.

Method 2: Pairing Terms

Group the series into pairs of $(1 - 1)$: $ (1 - 1) + (1 - 1) + (1 - 1) + \dots $ Each pair $(1 - 1)$ sums to 0.

With 101 terms, we can form $ \lfloor 101 / 2 \rfloor = 50 $ complete pairs. These 50 pairs account for $ 50 \times 2 = 100 $ terms.

The sum of the first 100 terms is $ 50 \times (1 - 1) = 50 \times 0 = 0 $.

The 101st term is the next term in the sequence, which is $+1$.

Therefore, the total sum is the sum of the first 100 terms plus the 101st term: $ \text{Sum} = 0 + 1 = 1 $.

The final result of the series calculation is 1.