Math, asked by nikhilkumishra3242, 1 day ago

Find maclaurin series of this question
F(x) = cos x

Answers

Answered by neetakarande79
0

Step-by-step explanation:

The Maclaurin series of

f

(

x

)

=

cos

x

is

f

(

x

)

=

n

=

0

(

1

)

n

x

2

n

(

2

n

)

!

.

Let us look at some details.

The Maclaurin series for

f

(

x

)

in general can be found by

f

(

x

)

=

n

=

0

f

(

n

)

(

0

)

n

!

x

n

Let us find the Maclaurin series for

f

(

x

)

=

cos

x

.

By taking the derivatives,

f

(

x

)

=

cos

x

f

(

0

)

=

cos

(

0

)

=

1

f

'

(

x

)

=

sin

x

f

'

(

0

)

=

sin

(

0

)

=

0

f

'

'

(

x

)

=

cos

x

f

'

'

(

0

)

=

cos

(

0

)

=

1

f

'

'

'

(

x

)

=

sin

x

f

'

'

'

(

0

)

=

sin

(

0

)

=

0

f

(

4

)

(

x

)

=

cos

x

f

(

4

)

(

0

)

=

cos

(

0

)

=

1

Since

f

(

x

)

=

f

(

4

)

(

x

)

, the cycle of

{

1

,

0

,

1

,

0

}

repeats itself.

So, we have the series

f

(

x

)

=

1

x

2

2

!

+

x

4

4

!

=

n

=

0

(

1

)

n

x

2

n

(

2

n

)

!

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