00:03
Okay, well to solve this exercise we're gonna use a trick.
00:07
That is we know that f of x is equal to l n of 1 plus 5 x and what is the derivative of this function? well the derivative of this function is f prime of x equals 5 multiplied by 1 over 1 plus 5 x and well we know that power series expansion of this function at zero is very easy to find.
00:38
That is, what do we have? this one can be written as 5 multiplied by 1 over 1 minus negative 5x like this.
00:52
And what is a power serious expansion of this function? well f prime of x can be written as 5 multiplied by a sum with k running from 0 up to infinity of negative 1 to the k multiplied by 5 to the k multiplied by x to the k multiplied by x to the k like this okay very well perfect now we know that the interval of compare of this power series here is negative 1 over 5 comma 1 over 5 now in order to find the mclaurin series of this function what can we do well we can integrate this power series term by term and so what are we gonna get well let's do it we are gonna have f of x which which is, okay, maybe let me just rewrite f prime of x like this, 5 to the k plus 1.
02:03
Okay, so it's gonna be, okay, k plus 1.
02:09
This way it's gonna be easier.
02:12
Okay, i was saying now let's integrate this power series term by term...