Approximate the value of the series to within an error of at...

Approximate the value of the series to within an error of at most 10^-4. $$sum_{n=1}^{infty} frac{(-1)^{n+1}}{n^6}$$ According to Equation (2): $$|S_N - S| le a_{N+1}$$ what is the smallest value of N that approximates S to within an error of at most 10^-4? N = 4 S ?

Transcript

00:01 Okay, we want to approximate the sum of this series to an error that's less than 10 to the minus four.

00:08 And we know that sn minus s, so if we truncate it at n, instead of going all the way to infinity, that's gonna be less than an plus one, where a is the individual term.

00:27 This is because the alternating series is zero, okay? so our error has to be 10 to the minus four.

00:46 We want 10 to the minus four to be less than, greater than an plus one, okay? so that means we got 10 to the minus four is greater than one over n plus 71 times n plus 80.

01:19 If we flip that over, it changes the direction of the inequality, so we get 10 to the fourth has to be less than n plus 71 times n plus 80.

01:34 So we need to find the smallest value of n that satisfies this inequality, okay? so this is n squared plus 151n plus 71 times eight, okay? 5680.

02:09 So we get, we have to satisfy this inequality.

02:13 So that means that n squared plus 151n has to be greater than 4320, because that's 10 ,000 minus this, because that's 10 to the fourth, okay? or n squared plus 151n minus 4320 is greater than zero, okay? so this is, this quadratic is, we can find the solutions of that, okay? and when we, if we just treat it equal to zero, and then those solutions will tell us the possible values of n.

03:40 There'll be fraction, you know, there'll be irrational numbers, but then we can determine from that what the smallest one is gonna be, the smallest integer that'll make this inequality satisfy.

03:58 So that tells us that n is minus 151 plus or minus the square root of 151 squared plus four times 4320 divided by two.

04:27 So here's our solutions.

04:29 So we don't want it to be negative, so we don't care about this one.

04:34 So this is the solution that matters.

04:37 So this is when it's equal to zero, okay? so for it to be greater than zero, it has to be bigger than this, okay? so that means that n equals 25...

Question

Approximate the value of the series to within an error of at most 10^-4. $$sum_{n=1}^{infty} frac{(-1)^{n+1}}{n^6}$$ According to Equation (2): $$|S_N - S| le a_{N+1}$$ what is the smallest value of N that approximates S to within an error of at most 10^-4? N = 4 S ?

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