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Approximate the sum of the series correct to four decimal places.

$ \displaystyle \sum_{n = 1}^{\infty} \frac {( - 1)^n}{(2n)!} $

$\approx-0.459722$

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Oregon State University

Baylor University

University of Michigan - Ann Arbor

Idaho State University

approximate the sum of the Siri's corrects of four decimal places. So here the first thing I observed is that it's alternating. Now. We want the ear to be less than or equal to five times ten to the negative Fifth. This is to ensure that were correct at the four decimal places. So in our case, since they here, since it's alternating, the ear is less than or equal to B in plus one. In our case, Bien for this problem is just one over two in factorial. This is just alternating Siri's estimation, Dirham, So we want be in plus one to be less than five times ten to the negative fifth. So equivalently one over two and plus one factorial less than five times ten to the minus five. Now, for example, we want to know how, what how large is and have to be to make this true. So let's go on to the next page and plug in some values event and people's too. Okay, one over two times two plus two factorial, That's one over sixty historial one over seven twenty. But this is Plug Goto a calculator. You could see that this is larger. This is too big for us. We do not want this value, men. So we try three. Let's go even further. Let's go for I'm sorry about that. I should not skip here. Got a little careless there. Let me actually go ahead and plug in three. So two times three plus two factorial, That's one over a factorial. And this one is already small enough for us. So this tells us that we can use three terms it take it, correct the four decimal places. So here the infinite sum over two in factorial is approximately equal to as three. So if you wanna plug that in plugin and equals one and equals two and an equal three and you'LL get thes first three fractions right here and simple Find this with a calculator. This is about point for five nine seven with a negative out there in the front. So that is correct for decimal places by the alternating Siri's estimation there. And that's our final answer