series expansion - List manipulation - various

OK, Please forgive the messiness of this, but I am working with something like:

d = (q = 8;
f = (lop = 
  Transpose[{Flatten[
     Reverse[Table[
        Gamma[y], {y, 1, q + 1}]] /. {ComplexInfinity -> 0}], 
    Flatten[Table[Binomial[n, k], {n, q, q}, {k, 0, n}]], 
    Flatten[Table[x^n, {n, 0, q}]]}];
 {#1*#2*#3} & @@@ lop);

Flatten[f];
Total[f]);
d

which gives:

{40320 + 40320 x + 20160 x^2 + 6720 x^3 + 1680 x^4 + 336 x^5 + 56 x^6 + 8 x^7 + x^8}

I am then comparing the roots with another series, eg:

Normal[Series[E^x, {x, 0, 8}]]

which, in this case are the same.

However, I have lots of different series & was wondering if I could manipulate them as stated below.

(NB - I realise there is no 'ComplexInfinity' output in the above example, but have left it in to demonstrate that I have tried several replace techniques for the subsequent questions.)

I have several questions on list manipulation & at the risk of asking the same question multiple times, I have compiled them into the following single question:

(a) Changing alternate (or some other pattern) signs / operators:

From this:

1 + x + (x^2)/2! +  (x^3)/3! + (x^4)/4! +  (x^5)/5! ...

I would like to do get this:

1 - x + (x^2)/2! -  (x^3)/3! + (x^4)/4! -  (x^5)/5! ...

(b) Multiplying alternate (or some other pattern) elements by a given value (eg - I):

From this:

1 + x + (x^2)/2! +  (x^3)/3! + (x^4)/4! +  (x^5)/5! ...

I would like to do get this:

1 - I x + (x^2)/2! -  I(x^3)/3! + (x^4)/4! -  I(x^5)/5! ...

(c) Convert all Imaginary values in a list to real values (or vice versa), noting that there may be no pattern in the distribution here:

From this:

1 + I x + (x^2)/2! +  (x^3)/3! + I(x^4)/4! +  I(x^5)/5! ...

I would like to do get this:

1 + x + (x^2)/2! +  (x^3)/3! + (x^4)/4! +  (x^5)/5! ...

or this:

I + I x + I(x^2)/2! +  I(x^3)/3! + I (x^4)/4! +  I(x^5)/5! ...

I have tried various select & replace methods, but have had no luck so far.

Answer

Well you can begin by wrapping your expression in Hold or HoldForm then using ReleaseHold when you want to Evaluate them. For your first example:

expr = HoldForm[1 + x + (x^2)/2! + (x^3)/3! + (x^4)/4! + (x^5)/5!]

Now you can do:

Replace[expr /. {Times[c : Power[x, n_?OddQ], d_] :> Times[-1, c, d]}, x -> - x, 2]

To get:

1 - x + x^2/2! - x^3/3! + x^4/4! - x^5/5!

For the next one:

Replace[expr /. {Times[c : Power[x, n_?OddQ], d_] :> Times[-I, c, d]}, x -> - I x, 2]

Which gives:

1 - I x + x^2/2! - (I x^3)/3! + x^4/4! - (I x^5)/5!

For the third:

expr2 = 1 + I x + (x^2)/2! + (x^3)/3! + I (x^4)/4! + I (x^5)/5!;

To change the imaginary numbers do:

expr2 /. Complex[0, c_] :> c

To change negatives to positives in e.g.

expr3 = -1 - x + (x^2)/2! - (x^3)/3! + (x^4)/4! - (x^5)/5!

Simply do:

expr3 /. {Times[-1, c_] :> Times[1, c], Rational[-1, d_] :> Rational[1, d], n_?Negative :> -1 n}

To get:

1 + x + x^2/2 + x^3/6 + x^4/24 + x^5/120