Limits of sequences practice problems
Does the sequence converge? The first few terms, then the limit as n → ∞. 10 problems with worked solutions; in 10 of them every equation is proved by a computer algebra system.
Write the first four terms of \( \displaystyle a_n = \frac{3 n^{2} + 1}{4 n^{2} + n} \), and determine whether the sequence converges. If it does, find its limit.
Write the first four terms of \( \displaystyle a_n = - n + \sqrt{n^{2} + 4 n} \), and determine whether the sequence converges. If it does, find its limit.
Write the first four terms of \( \displaystyle a_n = - n + \sqrt{n^{2} + 2 n} \), and determine whether the sequence converges. If it does, find its limit.
Write the first four terms of \( \displaystyle a_n = \frac{3 n^{2} - 3}{5 n^{2} + n} \), and determine whether the sequence converges. If it does, find its limit.
Write the first four terms of \( \displaystyle a_n = \frac{2 n + 4}{4 n^{2} + 1} \), and determine whether the sequence converges. If it does, find its limit.
Write the first four terms of \( \displaystyle a_n = \frac{n - 3}{3 n^{2} + 1} \), and determine whether the sequence converges. If it does, find its limit.
Write the first four terms of \( \displaystyle a_n = \operatorname{atan}{\left(n \right)} \), and determine whether the sequence converges. If it does, find its limit.
Write the first four terms of \( \displaystyle a_n = \frac{5^{n}}{n!} \), and determine whether the sequence converges. If it does, find its limit.
Write the first four terms of \( \displaystyle a_n = n \sin{\left(\frac{1}{n} \right)} \), and determine whether the sequence converges. If it does, find its limit.
Write the first four terms of \( \displaystyle a_n = \frac{\ln{\left(n \right)}}{n} \), and determine whether the sequence converges. If it does, find its limit.