= 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>:
$$
\frac{a^n}{b^n}=\left(\frac ab\right)^n.
$$
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,
$$
3\cdot2^n+5\cdot3^n\sim5\cdot3^n,\qquad
\frac{3\cdot2^n+5\cdot3^n}{5\cdot3^n}
=1+\frac35\left(\frac23\right)^n.
$$
The relative difference against $5\cdot3^n$ 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.
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