TITLE: Uniqueness of power series
QUESTION [6 upvotes]: Is there two sequences of real numbers $a_i$ and $b_i\neq 8$, not depending on $x$, such that $x^8=\sum_{k=1}^{\infty}a_kx^{b_k}$ for all $x$?
If $\displaystyle\sum_{k=1}^{\infty}a_kx^{b_k}=\sum_{k=1}^{\infty}c_kx^{d_k}$ for all $x>1$, and all coefficients are real and $b_k>b_{k+1}$, and $d_k>d_{k+1}$, is there a way to prove that $a_k=c_k$ and $b_k=d_k$ for all $k$?

If $\displaystyle\sum_{k=1}^{\infty}a_kx^{b_k}=0$ for all $x>1$ and $b_i\neq b_j$, and all coefficients are real, must $a_k=0$?

REPLY [2 votes]: Asked and answered at MO:
https://mathoverflow.net/questions/81482/uniqueness-of-power-series