Question

Determine if the following sequence converges or diverges: {(2n-1)/((3n^2)+1)} n= 1,2,3,.... If the sequence converges, find it's limit

Answer 1

Is this your \(a_n\)?
\[\large a_n=\frac{2n-1}{3n^2+1}\]
If so, then consider the function \(f(n)=a_n\) and find \(\lim_{n\rightarrow \infty} f(n)\) to get your answer.

Answer 2

the above response correctly assumes the correct a_n. I am confused on how to continue from there to check for convergence. My first thought was to use the ratio test but that doesnt seem to help much.

Answer 3

ratio test is for convergence of series, not just sequence

Answer 4

sequence convergence is just seeing if the terms tend to some value... take the limit:$$L=\lim_{n\to\infty}a_n=\lim_{n\to\infty}\frac{2n-1}{3n^2+1}$$the key is to consider that asymptotically we have \(2n-1\sim 2n\) and \(3n^2+1\sim 3n^2\) as \(n\to\infty\) (here \(\sim\) denotes asymptotic equivalence -- this boils down to them 'growing' identically in the long run). we can thus consider the nicer limit:$$L=\lim_{n\to\infty}\frac{2n-1}{3n^2+1}=\lim_{n\to\infty}\frac{2n}{3n^2}=\lim_{n\to\infty}\frac2{3n}=0$$so our sequence converges to \(0\)

Answer 5

alternatively you can justify the above using l'Hopital's rule or your 'tricks' for rational functions:$$\lim_{x\to\infty}\frac{a_0x^n+a_1x^{n-1}+a_2x^{n-2}+\dots}{b_0x^m+b_1x^{m-1}+b_2x^{m-2}+\dots}=\lim_{x\to\infty}\frac{a_0x^n}{b_0x^m}$$clearly if \(n=m\) our limit converges to \(a_0/b_0\); if \(n>m\) our sequence diverges off to infinity, and if \(n<m\) our sequence converges to \(0\)