Question

Prove that by Maclaurin's therorm:​

Answer 1

Step-by-step explanation:

We have:

f

(

x

)

=

x

sin

x

First let us take a look at the graph and see how "well defined" the function is:

graph{x/sinx [-40, 40, -20, 20]}

The Maclaurin series can be expressed in the following way:

f

(

x

)

=

f

(

0

)

+

f

'

(

0

)

1

!

x

+

f

'

'

(

0

)

2

!

x

2

+

f

'

'

'

(

0

)

3

!

x

3

+

(

f

(

4

)

)

0

4

!

x

4

+

...

=

∞

∑

n

=

0

f

(

n

)

(

0

)

n

!

x

n

We also note from the graph that

f

is even, so we expect all odd powers of

x

in the series to vanish. So, Let us find the derivatives, and compute the values at

x

=

0

.