00:02
I report this equation here.
00:05
The series is here.
00:06
The series here.
00:08
To prove certain relations that are given the option.
00:12
What we can do to prove a part, we can put x as negative 2.
00:17
They put here x as negative 2.
00:22
This part is coming out of a 0.
00:24
The whole series is coming out of the 0 here.
00:27
And let's put x here as negative 2.
00:30
So we get a0.
00:35
Then we have negative 2.
00:39
A1 then we have positive 2 square a 2 and so on we get negative 1 of the power n we have x negative 2 2 that's coming out to be a n times 2 to the power n so this be a report for the a part next what we can do we can now just take the same series here for the original equation and you multiply by x both sides so let's do that this series here we multiply by x both sides so we get here now to be x times x plus 1 x plus 2 and so on x plus n equal we get a n plus a plus a plus a plus a plus a plus x to part n plus 1.
02:06
Now we differentiate it with respect to x.
02:10
So let's differentiate it with respect to x.
02:12
So we get here a not plus 2 a 1x plus 3, a 2 x squared and so on, we get n plus 1, a n x2 par n.
02:28
That equals now.
02:30
Let's now differentiate this.
02:37
So so, taking the first term here, so the differential to act will be 1.
02:42
So we get x plus 1 times x plus 2 and so on till x plus n.
02:50
Plus using here the product rule.
02:53
The next we get here x, x plus 2, and so on we have x plus n because derivative of x plus 1 is 1.
03:07
And so on, we keep on derivating.
03:10
So we get it as term that's going to be x, x plus 1, x plus 2, and so on.
03:19
So we get a second last term, x plus and negative 1.
03:22
Derivative of this is coming out to be, or we can say the original series, derivative is coming out to be.
03:30
So we get this equation.
03:36
Now in this, what you can do here, for in b part, we can put x is equal to 1.
03:41
Now, if we put x equal to 1, we get it as a notch plus 2a1 plus 3a2 and so on, we have n plus 1, and this equals we'll get here, it's coming out to be 2 times 2 and so on.
04:17
We get the input here, which you can here access 1.
04:23
So this part is coming up to be 3.
04:25
And this part is coming out to be here.
04:31
Axis 1 will become 1 plus.
04:36
Next we get 1 times 3 and so on.
04:41
1 plus n.
04:43
Continue like this.
04:45
So we get 1 times 2 times 3 and so on.
04:51
We have n here.
04:56
So we get now, this is a0 plus 2a1 plus 3a2 and so on.
05:05
N plus 1 a and that equals this series here for sequence x coming out to be n plus 1 factorial this will be we multiply and divide by 2 it will become n plus 1 factorial by 2 so we get n plus 1 factorial factorial plus n plus 1 factorial by 2 to be 2 here and so on will become n plus 1 factorial by n plus 1 so this part here we multiply and divide by n plus 1 we go this n plus 1 factorial by n plus 1 and this coming out to be here n plus 1 factorial we get 1 plus 1 over 2 and so on 1 over n plus 1 next for c option what we can do we can do here for the original series here we can replace x by 1 over x and then we multiply by extra power n.
06:19
For example, if i do this here, we just play this as with 1 over x, it will become a1 over x.
06:33
Play it as a1 over x.
06:36
And i multiply this with extra power n.
06:40
I'm just doing it up for the one term only.
06:43
So this is having out to be a1, extra power and negative 1.
06:47
Similarly here on the left -hand side of this multiply with x to par n.
06:51
So first it replaced with 1 over x.
06:55
This will become here 1 plus 1 over x.
06:58
I'm just doing it here on the 1 on the first term only...