Calculus I · Introduction to sequences and series
Diagnose a false convergence claim from the term limit
Problem
Consider \[ \sum_{n=1}^{\infty}\frac{2n+1}{3n-2}. \] A student argues: “The terms approach $2/3$, which is less than $1$, so the series converges.” Diagnose every flaw in this argument and determine whether the series converges or diverges. State precisely what the nth-term test does and does not say.
Hint
First compute the limit of the term by dividing numerator and denominator by $n$.
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