Taylor Series Help: ƒ(x) = e^(x/2), g(x) = ((e^(x/2)) - 1)/x

[thedudescousin]

Let ƒ be the function given by f (x) = e ^ (x / 2)

(a) Write the first four nonzero terms and the general term for the Taylor series expansion of ƒ(x) about x = 0.
(b) Use the result from part (a) to write the first three nonzero terms and the general term of the series expansion about x = 0 for g (x) = ((e^(x / 2)) – 1)/x

For part a I got
P4 = 1 + x/2 + ((x/2)^2)/2! + ((x/2)^3)/3!

E ((x/2)^n)/n!
E = summation sign n=0 to infinity

For part b I got, which was marked wrong,
x-1 + ((x/2)2 – 1)/(2!x) + ((x/2)3 – 1)/(3!x)

E ((x/2)n - 1)/(n!x)

On my paper my teacher marked only –1 once and I am not for sure what that means. Any help on part B would be greatly appreciated.

 

[HallsofIvy]

Let ƒ be the function given by f (x) = e ^ (x / 2)

(a) Write the first four nonzero terms and the general term for the Taylor series expansion of ƒ(x) about x = 0.
(b) Use the result from part (a) to write the first three nonzero terms and the general term of the series expansion about x = 0 for g (x) = ((e^(x / 2)) – 1)/x

For part a I got
P4 = 1 + x/2 + ((x/2)^2)/2! + ((x/2)^3)/3!

E ((x/2)^n)/n!
E = summation sign n=0 to infinity

Good!

For part b I got, which was marked wrong,
x-1 + ((x/2)2 – 1)/(2!x) + ((x/2)3 – 1)/(3!x)

E ((x/2)n - 1)/(n!x)

On my paper my teacher marked only –1 once and I am not for sure what that means. Any help on part B would be greatly appreciated.

You were, of course, correct that the Taylor's series for e^(x2) is 1+ x/2+ ((x/2)^2)/2!+ ((x/2)^3)/3!+ ...
But for (b) you seem to have just stuck a "-1" into the numerator, and divided some of the denominators by x. Do it step by step:
e^(x/2)= 1+ (x/2)+ (x/2)^2/2!+ (x/2)^3/3!+ ... (which I would prefer to write as 1+ x/2+ x^2/((4)(2!))+ x^3/((8)(3!))+ ...) so
e^(x/2)-1= x/2+ x^2/((4)(2!))+ x^3/((8)(3!))+ ... and finally
(e^(x/2)-1)/x=(1/2)+ x/((4)(2!)+ x^2/(8)(3!)+...
I think that is
$$\sum_{n= 0}^\infty \frac{x^n}{2^{n+1}(n+1)!}$$

 

[thedudescousin]

Thank you, it actually makes sense now.

 

[hereiscassie]

good!

there was a part c for this too. it said:
c) for the function g in part b, find g'(2) and use it to show that $$\sum_{n= 1}^\infty \frac{n}{4(n+1)!}$$ = 1/4
i don't know how to start it, or what to do. help please?