Mathematics

Mathematics

Problem

A sequence is defined recursively by a1=2a_1=2 and an+1=3an+14a_{n+1}=\dfrac{3a_n+1}{4} for n≥1n\ge 1. Using the behavior of ana_n as n→∞n\to\infty, determine whether the series ∑n=1∞an\displaystyle \sum_{n=1}^{\infty} a_n converges or diverges.
Intuitive explanation

The key is to find what the sequence approaches as n→∞n \to \infty, then use that limit to determine the series' fate. If ana_n converges to a nonzero value, the terms don't shrink to zero, so the series must diverge by the divergence test. Set up the recurrence as a limit equation to find that fixed point, then check whether the series can possibly converge.

Intuitive explanation

The key is to find what the sequence approaches as n→∞n \to \infty, then use that limit to determine the series' fate. If ana_n converges to a nonzero value, the terms don't shrink to zero, so the series must diverge by the divergence test. Set up the recurrence as a limit equation to find that fixed point, then check whether the series can possibly converge.

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