Determine if the following sequences converge or diverge.  If the sequence converges find out its limit.

a. {3n² - 1 / 10n + 5n²} ^∞ _{n =2}

b. {e²ⁿ / n}^∞ _{n =1}

c. {(-1)ⁿ / n}^∞ _{n =1}

d. {(-1)ⁿ}^∞ _{n =0}

Solution

(a) {3n² - 1 / 10n + 5n²} ^∞ _{n =2}

In the above case all we need to do is remind the method that was developed earlier to deal with the limits of rational functions. 

 To do a limit in this type all we require to do is factor from the numerator and denominator the largest power of n, cancel and after that take the limit.

Thus the sequence converges and its limit is 3/5.