For the sequence defined by \(a_n=\frac{3n}{n+1}\) for integers \(n\ge 1\), find \(a_1\), \(a_2\), \(a_3\), and \(\lim_{n\to\infty}a_n\).

For the geometric sequence \(a_n=5\left(\frac{1}{2}\right)^{n-1}\) for integers \(n\ge 1\), find the first four terms and the limit as \(n\to\infty\).

A sequence satisfies \(a_1=2\) and \(a_{n+1}=a_n+3\) for integers \(n\ge 1\). Find the first four terms and an explicit formula for \(a_n\).

Show that the sequence \(a_n=1-\frac{1}{n}\), defined for integers \(n\ge 1\), is increasing and bounded above by \(1\).

Find \(\lim_{n\to\infty}\frac{2n^2+1}{5n^2-3}\).

Find \(\lim_{n\to\infty}(\sqrt{n^2+n}-n)\).

Determine whether the sequence \(a_n=\frac{(-1)^n}{n}\), defined for integers \(n\ge 1\), converges. If the sequence converges, find the limit.

Determine whether the sequence \(a_n=(-1)^n\), defined for integers \(n\ge 1\), converges.

Find the smallest integer \(N\) such that \(\left|\frac{1}{n}\right|<0.01\) for every integer \(n\ge N\).

A learner claims \(a_n=\frac{n+1}{n}\) diverges because the numerator and denominator both grow without bound. Explain the error and find the limit.