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Equences and SeriesEvaluate:X6-1 5(2)n-1 = 2 |Enteralccllzs Corporutin Af Righis Reccredcef...

Question

Equences and SeriesEvaluate:X6-1 5(2)n-1 = 2 |Enteralccllzs Corporutin Af Righis Reccredcef

equences and Series Evaluate: X6-1 5(2)n-1 = 2 | Enter alccllzs Corporutin Af Righis Reccred cef



Answers

Expand and evaluate each series. $$\sum_{i=2}^{4} i^{2}$$

Okay, so for this example we are given E to the X. Sign of acts and were given that A is equal to pi over two. And we need to find the taylor series expansion for our function E to the X. Sine of X. So we know that that taylor series expansion for a function is the function evaluated at A. Plus. It's derivative, evaluate A. A. Multiply by X minus A. Plus. It's second derivative evaluator, A multiplied by X minus a squared over two factorial plus. It's third derivative evaluator A multiplied by X minus a cubed over three factorial plus some additional terms. Okay, so our function is F. Of X is equal to E to the X. Sine of X. And we need to first find the function evaluated at A. And A is pi over two. Okay, so therefore F. Of A. Is equal to E. To the pi over two. Multiplied by sign of pi over two. And this is just equal to one. So then we get E to the pi over two. Okay? So now ah let's just find our sorry our next piece here. So we need to find our first derivative evaluated A. So F prime of X is equal to. So we'll have a product here. So that will give us E to the X. There will be co signed of X plus sign of X. Okay. And now we need to evaluate our derivative at hi over two. Okay. So let's see. Co sign a pi over two. This is zero and this is one. So therefore we'll get E to the hi over to. All right. So now we need to find our second derivative here. Okay, So F double prime of X. Is equal to. So let's see. We'll have E to the acts and then we'll have minus sine of X plus co sign of eggs. Close co sign of X plus sine of X. Okay. And the signs are going to cancel. So then we just have to ah he to the X. Co sign of X. Okay. So now we need to evaluate our second derivative that are at A. So we'll have zero because co sign A pi over two is zero. So therefore her second derivative will be zero. All right now let's find our third derivative. Okay, so F triple prime of A is equal to. So let's see we will have the O. P. Two. He to the acts. Then we'll have co sign of acts minus sine of X. Okay. And now we just need to evaluate this other point. Uh So we'll have that F triple prime of A is equal to. So that will be uh minus two E. To the hi over to. All right, well, we have everything that we need. We have all of our derivatives. So let's just go ahead and make her serious. So then we have E to the X sign of X is equal to. So we'll have E to the pi over two uh plus each of the pi over two. X minus pi over two. And then for our second derivative we have zero. So then we'll have minus. And ah if you recall from our expansion formula, we have three factorial, three factorial Z equals six. So we have a two on the top. So they'll just simplify to one third E to the pi over two, X minus pi over two Q plus some additional terms. Now we do have a factor of E to the pi over two. So let's just bring that out. So E to the X. Sine of X is equal to E. To the pi over two, one plus X minus pi over two minus one third, X minus pi over two cube plus some additional terms. And this is the answer.

Okay, so let's find the falling Siri's. Let's start with eyes equal to one. So that's one squared. So it's one of two and all its white with I is equal to two. So that's to score incidents for over two. And then for Isaac to three. We have three Squared, which is nine over to, and then for our fourth turn. So it's force words that at 16/2 Okay, let's get the fallen. And now it's under falling. Okay, so that gives us 15.

The city is information. I want three. I was here last two. But this isn't quarto Won't scare last two less pool scared. Blessed to Yes, to be used here. The one scary one went must with three. It was getting bored or Mr With six. He scared his 99 Mr Lose 11. So when we are there, these numbers we will have reverse it is 99 and 11 years.

This question. Your final video distribution. Who blessed by Uranus? I swear. I was to warn you soon. You get any numbness? Well, this formation I six this summation. I didn't do six. Wariness. Permission, I swear. What is No, the really of this PB submission one, maybe 81 Movements extend to this big toe. Six. Bless. This is some of the first six days of us. There could be a bigger for 16 to 70. Every two minus two CV submission. All the first six naturally square some of this six and natural. Most of the solution. All directors but some of their this and cancels a cell and listening one plus one. So there nothing They are 0666 grayness, maybe one. So there are commuting. Doesn't need me. Miss your Be OK, So this is nonsense issue for his question.


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