calculus question help please

Question

Show that the series  is convergent by the alternating series test, and find the number of terms necessary to estimate the sum of the series with an error of less than 0.05.

Answer 1

This is ∑(-1)ⁿ⁺¹1/n², so b_n = 1/n² and b_n is positive and decreasing for all n.

If lim n→∞ b_n = 0 then the sum converges. b_n does converge to 0

|a_n+1| ≤ 0.05

1/(n+1)² ≤ 0.05

(n+1) ≥ √20

n ≥ 3.472 ≈ 4

Need at least 4 terms.

Checking:

n |1/(n+1)²|

1 0.5

2 0.111111

3 0.0625

4 0.04 < 0.05

Answer 2

The alternating series test says that if the terms in the series alternate signs, then it is sufficient to show that the terms themselves converge to 0 for the series to converge. (Note: this is a necessary but not sufficient condition for a series of strictly positive terms to converge. For example, the harmonic series, a_n = 1/n is divergent even though lim_n→∞ a_n = 0.)

The generating sequence here is a_n = (-1)ⁿ⁺¹ / n².

lim_n→∞ (-1)ⁿ⁺¹ / n² = 0 and so the series converges.

(Btw, the series converges absolutely because it is a p-series with p = 2 > 1 , and any alternating series that converges absolutely is convergent.)