Limit Comparison Test Solved Examples1.Determine whether the series converges or diverges: \(\sum_{n=1}^{\infty}\frac{(2n^2 + n)}{(3n^4 + 5)}\).Solution:First, we need to choose a comparison series. Let's try the series sum from \(n = 1\) to infinity of \(\frac{1}{n^{2}}\). This is a \(p\)-series with \(p = 2\), which we know converges.Now, we take the limit of the ratio of the two series as \(n\) approaches infinity:\(\displaystyle \lim_{n\to\infty}\frac{\frac{(2n^{2}+n)}{(3n^{4}+5)}}{\frac{1}{n^{2}}}\)\(=\displaystyle \lim_{n\to\infty}\frac{(2n^{2}+n)}{(3n^{4}+5)}\times\frac{n^{2}}{1}\)\(=\displaystyle \lim_{n\to\infty}\frac{2+\frac{1}{n}}{3+\frac{5}{n^{4}}}\)\(=\frac{2+0}{3+0}\)\(=\frac{2}{3}\)Since this limit is finite and positive, and the series we chose to compare it to converges, we can conclude that the original series also converges.2.Determine whether the series converges or diverges: \(\sum_{n=1}^{\infty}\frac{(3n+2)}{(n^{2}+1)}\).Solution:Again, we need to choose a compari...
