00:01
In this problem, we have two parts.
00:04
So let's get started with the first one.
00:06
In the first part, we will find the maclaurin series representation for the function f of x equal to exponential minus x to the power 5.
00:17
So remembering that exponential y, let's say, is equal to summation from k equal to 0 to infinity 1 over k factorial y to the power k.
00:27
We can directly obtain this series representation by replacing y by minus x to the power 5.
00:40
So we have summation 1 over k factorial minus x to the power 5 to the power k.
00:48
So we have summation from k equal to 0 to infinity minus 1 to the power k divided by k factorial x to the power 5k.
01:00
This is the answer.
01:03
In the second part, we are going to use this series to approximate this interval.
01:10
Let's call it i.
01:11
From 0 to 0 .5 exponential minus x to the power 5 dx with an error.
01:20
Let's call the error epsilon less than or equal to 0 .000003.
01:30
We have for this interval from 0 to 0 .5 dx summation from k equal to 0 to infinity minus 1 to the power k over k factorial x to the power 5k.
01:46
Bringing all the k dependent terms out, we have summation minus 1 to the power k over k factorial integral from 0 to 0 .5 dx x to the power 5k.
02:04
So we have summation from k equal to 0 to infinity minus 1 to the power k over k factorial 0 .5 to the power 5k plus 1 over 5k plus 1...