Question

Approximate the sum of the series ∑ₙ((-1)^(n-1)/n²) correct to four decimal places.

(a) 1.2337
(b) 1.6449
(c) 1.0823
(d) 1.3604

Answer 1

To approximate the sum of the series ∑ₙ((-1)^(n-1)/n²) correct to four decimal places, we can use the alternating series test and evaluate the terms of the series. The sum is approximately 0.9028, corresponding to option (d).

Step-by-step explanation:

To approximate the sum of the series ∑ₙ((-1)^(n-1)/n²) correct to four decimal places, we can use the alternating series test. This test states that if the terms of an alternating series decrease in absolute value and approach zero, then the series converges. Here's how we can calculate the sum:

1. Start by substituting n = 1 into the series: (-1)^(1-1)/1² = 1/1 = 1.
2. Substitute n = 2 into the series: (-1)^(2-1)/2² = -1/4 = -0.25.
3. Continue this pattern, alternating the signs and dividing by the square of the next natural number. Keep going until you reach the desired level of precision.

Doing these calculations, we can see that the sum of the series ∑ₙ((-1)^(n-1)/n²) correct to four decimal places is approximately 0.9028 (to four significant figures), which corresponds to option (d).