Question 12(3 points)

By the binomial series formula, we find1

1+x

=(1+x)

1/2

=11

2

x+3

8

x

2

...

{"version":"1.1","math":"\(\frac{1}{\sqrt{1+x}=(1+x)^{-1/2}=1-\frac{1}{2}x+\frac{3}{8}x^2-...\)"}.

The Maclaurin series of the functionf(x) =ln(x+4+x

2

)

{"version":"1.1","math":"\(\ln{(x+\sqrt{4+x^2})}\)"}can be found by this binomial series and the fact thatd

dx

ln(x+4+x

2

)=1

4+x

2

{"version":"1.1","math":"\(\frac{d}{dx}\ln{(x+\sqrt{4+x^2})}=\frac{1}{\sqrt{4+x^2}\)"}.Let the Maclaurin series beln(x+4+x

2

)=

n=0

c

n

x

n

{"version":"1.1","math":"\(\ln{(x+\sqrt{4+x^2})}=\sum_{n=0}^\infty c_nx^n\)"}.Thenc

5

{"version":"1.1","math":"$c_5$"}=

Question 12 options:

a)

3

8

{"version":"1.1","math":"\(\frac{3}{8}\)"}

b)

3

1280

{"version":"1.1","math":"\(\frac{3}{1280}\)"}

c)

3

1280

{"version":"1.1","math":"\(-\frac{3}{1280}\)"}

d)

1

48

{"version":"1.1","math":"\(-\frac{1}{48}\)"}

e)

1

3240

{"version":"1.1","math":"\(\frac{1}{3240}\)"}

f)

1

48

Question 12 (3 points) By the binomial series formula, we find -= (1 + x) 2 = 1 - 2x+ 2x2-.... The Maclaurin series of the function f (x) =In (x + V4 + x2) can be found by this binomial series and the fact that & In (x + V4 + x2) = V 4+ x 2 Let the Maclaurin series be In (x + V4+ x2) = En-o Cnx". Then C5 = O a) OO / W Ob) 3 1280 O c _ 3 1280 O d) _ 1 48 Oe) 1 3240 Of) 1 48