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I came across a sequence of the form $|4^n + (-3)^n|^{1/n}$ and was unsure where to start in terms of computing its limit. Is there a general way to approach sequences of this form with $1/n$ in the exponent? Without brute force computation.

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  • Have you tried using logarithms? Usually, logarithms + some inequality bashing helps with finding the limits of these kinds of sequences. – Mousedorff Oct 27 '22 at 17:26
  • See: https://math.stackexchange.com/q/484451/42969, https://math.stackexchange.com/q/777377/42969, https://math.stackexchange.com/q/923219/42969, https://math.stackexchange.com/q/73834/42969 – Martin R Oct 27 '22 at 17:27
  • Also this https://math.stackexchange.com/q/3084056/42969 and the linked questions. – Martin R Oct 27 '22 at 17:30

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Hint

$$\bigg|4^n + (-3)^n\bigg|^{1/n} = \bigg|4^n\left(1 + \dfrac{(-3)^n}{4^n}\right)\bigg|^{1/n} = 4\cdot \bigg|1 + \dfrac{(-3)^n}{4^n}\bigg|^{1/n}$$