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JH
Numerade Educator

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Problem 23 Easy Difficulty

Show that the series is convergent. How many terms of the series do we need to add in order to find the sum to the indicated accuracy?

$ \displaystyle \sum_{n = 1}^{\infty} \frac{( - 1)^{n+1}}{n^6} (|error| < 0.00005) $

Answer

since the 6 th term is less than the desired error, we need to add the first 5 terms to get the sum to the desired accuracy.

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Video Transcript

first, let's show that the series is conversion. The series is alternating. So let's to find Bien to be won over and six. This is the positive part of the A N here now, First of all, we see that this is positive. So we're using the alternating Siri's test or serum with the book calls it. So there's some conditions. First, we have to have that Beyonce positive sense and is bigger than one one over into the six is always positive. That's the first condition. Second one, I need the limit to be zero. That's clearly true because the nominator goes to infinity. The numerator is just one, and usually the heartbroken listen to check in this case, it's not bad at all is that it's decreasing. This is true because in our problem, that's saying, if you increase and buy one, does the fraction get smaller? We could clearly see that that is true because the denominator on the left is larger. Therefore, the Siri's conversions by the alternating Siri's there. Um, that's just the first part of the question. There's some more work to do because now we'd like to know how many terms do we have to add. That's the end value, the number of terms and the partial. Some needed to ensure that the ear is less than point zero zero zero zero five. Let's go to the next page to work on this, so we would like to know when be end is less than point zero zero zero zero five. So here we just go to the calculator and compute these one at a time until we get one that works. The one is just one too large, still too large, and you we would keep going in this direction the first time that I come across. One that's small enough is that when an equal six, So this is Mohr. Less negative sees me here. I'm taking up flew value or not, I'm looking at being point zero zero zero zero two one, and that's smaller than the point zero zero zero zero five that we asked for. Therefore, we're using the fact that the air is bounded above by being plus one, and this we just showed that this is less than point zero zero zero zero five. If the sub script and plus one is six or larger, so This means that we need end to be bigger than her equals O five in the number of terms. So I need five terms in the Somme to ensure that the ear is small enough. So five terms and that's our final answer.