 Asymptotic comparison of exponential functions

For positive bases $a,b$, comparison of the exponential functions $a^n$ and $b^n$ reduces to a geometric progression:If $a<b$, this ratio tends to zero, so $a^n=o(b^n)$ in little o notation. If $a=b$, the ratio is one; if $a>b$, it grows without bound.

Consequently, provided at least one coefficient is nonzero, a finite sum with distinct positive bases is asymptotically equivalent to its largest-base term whose coefficient is nonzero. For example,The relative difference against $5\cdot 3^n$ first drops below one percent at the integer $n=11$. Dividing by the proposed dominant term provides an explicit error estimate, not just a comparison of two rapidly growing numbers. If several terms have the same largest base, their coefficients must first be combined; cancellation can remove that base entirely.