Is there a way to simplify this?

Question

1-x+x^2-x^3+x^4...+x^2n

Answer 1

Well, it depends on what you mean by simplified (it cannot be factored, and terms cannot be combined any further).However,
if you don't want the ellipsis, you can write it as:

[x^(2n+1) + 1]
────────
x + 1
for x ≠1

{ That is, [x^(2n+1) + 1] / (x+1) }

since (x+1) * (1-x+x^2-x^3+x^4...+x^2n) = x^(2n+1) + 1
It is more compact, but x+1 can be canceled out of both numerator and denominator. (also, it is not defined for x=-1 whereas your form is.)

Answer 2

Let E = 1 - x + x^2 - x^3 + x^4 - ... - x^(2n - 1) + x^(2n)

Multiply through by x.
xE = x - x^2 + x^3 - x^4 + x^5 - ... - x^(2n) + x^(2n + 1)

Add the 2 equations together.

E + xE = 1 + x^(2n + 1)
E(1 + x) = 1 + x^(2n + 1)

Therefore, E = [ 1 + x^(2n + 1) ] / (1 + x)
provided x ≠ 1.

Answer 3

Sum, i = 1 to 2n, of (-x)^i.

In TI-BASIC, Sum(i, 1, 2n, (-x)^i)

On paper, write a capital Sigma. This looks like a capital M, rotated 90 degrees counterclockwise so as to resemble an E. Above it, write 2n. Below it, "i = 1". To the right, (-x)^i.

Note that this is only an abbreviation, not a simplification. Unfortunately, I don't think a real simplification exists.

Answer 4

this is the summation of (-x)^t...from t=0 to t=2n

Answer 5

try getting rid of them there letters and put some real numbers in there!