12 HL The Ratio Test

Complete the following questions before our next class.

Use the ratio test to determine the convergence or divergence of the following series.

  1. \[\sum_{n=1}^\infty \frac{n^3 3^{n+1}}{4^n}\]
  2. \[\sum_{n=1}^\infty \frac{n^{10}}{10^n}\]
  3. \[\sum_{n=1}^\infty \frac{n^n}{n!}\]
Click for a hint to question 3.
Consider the definition of \(e\).

From the Haese book, complete Exercise K.3 question 1.

12 HL Sequences and Convergence

Complete the following questions before our next class.

(From the Haese book)
Exercise B.1 questions 3 and 5
(Read Section J, then complete )
Exercise J.1 questions 1bdf, 2acdf, 3
Exercise J.2 questions 1, 2, 3

Pay particular attention to questions 1 and 2 in J.2, which makes use of two (often useful) algebraic methods for showing that a sequence is monotonically increasing/decreasing: a sequence is decreasing if either \(u_{n+1}-u_n <0\), or \(\frac{u_{n+1}}{u_n} <1\) for all \(n\), and with inequalities reversed either method establishes that the sequence is increasing.