Say you had the sequence $U_{n+1} = 2bU_n$ where $U_1 = 6$. How would you find the range of values of b for which the sequence converges?
2026-05-05 02:22:41.1777947761
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How to find range of values for which a sequence converges?
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The sequence $U^{n+1} = 2bU^n$, $U^1 = 6$, is a geometric progression with ratio $2b$ and first term $6$. The general term is $U^n = 6(2b)^{n-1}$. The sequence is convergent for ratio $1 < 2b \leq 1$ and $$ \lim_{n \to \infty} U^n = \begin{cases} 0, \quad |2b|<1, \\ 6, \quad 2b = 1. \end{cases} $$
Essentially, you have to find out for which values of x a sequence x^n converges. These are all values, which converge to zero (don't miss the negative ones!) and one more special case.