Final answer:
To determine the number of terms needed to ensure that the remainder is less than 10^-5 in magnitude for the given convergent series, we need to use the Remainder Estimation Theorem for Alternating Series. This theorem states that the remainder after summing the first n terms is less than or equal to the absolute value of the (n+1)th term. By solving the inequality |(-1)ⁿ⁺¹/(7(n+1)+5)²| < 10^-5, we can determine the minimum value of n that satisfies the inequality and ensures the desired level of accuracy.
Step-by-step explanation:
To determine how many terms of the given series must be summed to be sure that the remainder is less than 10-5 in magnitude, we need to use the Remainder Estimation Theorem for Alternating Series. This theorem states that the remainder Rn, after summing the first n terms of an alternating series, is less than or equal to the absolute value of the (n+1)th term.
The given series is ∑[infinity]ₖ₌₀ (-1)ᵏ/(7k+5)². We need to find the value of n such that |(-1)ⁿ⁺¹/(7(n+1)+5)²| < 10-5. We can do this by solving the inequality.
Let's solve the inequality:
|(-1)ⁿ⁺¹/(7(n+1)+5)²| < 10-5
From here, we simplify and solve for n:
1/(7(n+1)+5)² < 10-5
(7(n+1)+5)² > 10⁵
49(n+1)² + 70(n+1) + 25 > 10⁵
Now, we can solve this quadratic inequality to find the range of n that satisfies it. Once we find the range, we can determine the minimum value of n that ensures the remainder is less than 10-5.