Write the Power Series expression for a given sequence

[Altagyam]

Homework Statement

http://sites.math.rutgers.edu/~ds965/temp.pdf (NUMBER 2)[/B]

Homework Equations

I do not understand the alternating part for the second problem and the recursive part for the first problem.

The Attempt at a Solution

The first answer I got was first by writing out the general expression for a power series and realizing that the center must be zero. Thus I wrote:
Σ w/ n=0 to n=100 (a_n)xⁿ

However for the second part I don't really understand how to do the problem. The sequence portion only reaches n=50 while x=100. I thought to myself that "Okay, then n for x must increase by a factor of 2n and there is something going on with the sequence that only let's it reach 50. Now my issue is, I don't understand how or where to apply this hint given.

Thanks in advance.

 

[bolometer]

Hi.
The 2nd problem doesn't actually ask for a series expression.

The key point here is to understand how the term $f^{(n)}(0)$ will look like. Think what the derivation of order n will do to every term of the function.

If you have understood that, then you'll see how to write the "concise expression" required.

Same thing for the g function, it's only a bit trickier.

 

[Altagyam]

Hi.
The 2nd problem doesn't actually ask for a series expression.

The key point here is to understand how the term $f^{(n)}(0)$ will look like. Think what the derivation of order n will do to every term of the function.

If you have understood that, then you'll see how to write the "concise expression" required.

Same thing for the g function, it's only a bit trickier.

Am I right for the first question though? We literally just covered this and I am still scratchy on this.

 

[bolometer]

$\Sigma _ {n=1}^{100} a_n x^n$
Is not the right answer.
As I sad

The 2nd problem doesn't actually ask for a series expression.

Ditch that line of action, take a breath and think it out, step by step.Hint:Please try to use/learn latex imput, it really makes math better. Chances are you'll need it anyway. ;)