In Exercises 29–42, find each indicated sum. 4Σi=0 (−1)^i/i!

Question

Accepted Answer

Hello. Today we are asked to evaluate the indicated summation. So what we are given is the summation from X is equal to 0 to 3 of negative two to the power of two X over X factorial. Now, in order to evaluate the indicated summation we need to first figure out what values of X. We're going to plug into our pattern. The starting value of X can be found at the bottom of the sigma symbol and that's going to be X is equal to zero. So our starting value is going to be X is equal to zero. And now we need to take a look at what value we're going to be calculating up to the final value for X could be found at the top of sigma and that's going to be three. So we're gonna be plugging in all the values starting at zero and going all the way to three. So there's a total of four values of X that we're gonna be plugging into our pattern. Now what we want to go ahead and do is take each value of X plug into the pattern to get one of the sums or one of the terms of the sum and add up all the terms together. So in order to start that process let's go ahead and plug in the value of X. When X is equal to zero. When X is equal to zero we get negative two to the power of two times zero. All over zero factorial zero factorial is equal to one. So you can go ahead and replace the denominator with one And on the numerator, the exponent is two times zero which is just going to give us zero. So we have negative two to the power of zero. Anything to the power of zero is just going to give us one. So we have negative one times one which is just going to give us negative 1/1. And that means that the first term of the sum is going to be negative one. Now we want to go ahead and repeat this process for the remaining values of the index. So next we plug in the value X is equal to one and that's going to give us negative two to the power of two times 1/1 factorial one factorial is just going to give us one. So we have one in the denominator and on the numerator, the expo in two times one is just going to give us two. So what we have is negative two to the power of two and two to the power of two is going to give us four. So we have negative 4/1 which is going to give us negative four and that's going to be the next term in the some Next we plug in the value X is equal to two and when X is equal to two we get negative two to the power of two times 2/2 factorial. Now in the numerator, two to the power to the power of two times two is going to give us two to the power of four. So we have negative two to the power of 4/2 factorial which is two times one. Two times one is just going to give us two and two to the power of four is going to give us 16. So what we are left with is negative 16/ and that's going to be negative eight which is the next term And finally we plug in X is equal to three which is going to give us negative two to the power of two times three over three factorial. Now in the Denominator three factorial can expand to three times 2 times one. And in the numerator two to the power of two times three is going to give us two to the power of six. So what we have is negative two to the power of six. Three times two times one is just going to give us six and two to the power of six is going to give us 64. So we have negative 64/6 which simplifies to negative 32/3 and that's going to be the final term of the sum. Now again we're going to take the four terms that we calculated and put them together as a single some and that's going to give us negative one minus four minus eight -32/3. And adding these values together is going to give us -71/3, which is going to be the total sum of the given summation. And with that being said, the answer to this problem is going to be a so I hope this video helps you in understanding how to evaluate a summation, and I'll go ahead and see while in the next video.

In Exercises 29–42, find each indicated sum. 4Σi=0 (−1)^i/i!

Key Concepts

Summation Notation

Factorial

Alternating Series

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