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JH
Numerade Educator

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Problem 26 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} \left( - \frac {1}{n} \right )^n (|error| < 0.00005) $

Answer

We only need to add the first five terms of the series to approximate the sum within the allotted error.

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Video Transcript

first, let's show that the Siri's is conversion, so but he's the alternating Siri's test. So in this case, we should take BNC, just be won over entity and our, and this easily satisfies that all the conditions on one hand, we can see that this is always positive because numerator and denominator positive the limit of bien. So this is the second condition is zero because the denominator those two infinity numerator is just one. And lastly, bien plus one is less than or equal to be in this ends if we just replace. And with n Plus one, that denominator gets larger, but as a whole, the fracturing it smaller. So that's true, too, for all in. So the Siri's converges by the illustrating Siri's their own being satisfies all three hypotheses, and that's all that's needed. Now there's a second part of the question, so it's maybe switch color here too blue. How many terms of the Siri's do we need to add to ensure a some of the indicated accuracy? So here we just would like to know how far how big should end be so that if we approximate the some using this sn that the approximate here is on. Ly is less than four zeros, followed by a five after the decimal. So here will use the alternating Siri's estimation. Dirham. So this is also an eleven point five. So this is that the ear. So I'll go to the next page here. This serum says that they're here absolute value when using here This is for when using in terms to approximate the infinite sum This is less than bien plus one and we want that to be less than point zero zero. So four zeros after decimal followed by a five. So let's just go ahead and solve this inequality for N So this is equivalent to and soda and bigger than twenty thousand and the smallest end outworks hereafter do It's in trial and error. Six. However, this really we should have used and plus one because this is the index on the be so really we want and plus one to be better than six. But that just means and is bigger than her equals o five. Because again, notice the difference here on the sub scripts. Five terms corresponds to be six. So this is why I will take five terms in order to ensure that the error is less than this number off here. So five terms and that's the final answer.