Science:Math Exam Resources/Courses/MATH103/April 2017/Question 01 (a)

MATH103 April 2017

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Question 01 (a)
Given the following general terms $a_{n}$, determine whether the corresponding sequences $\{a_{n}\}_{n\geq 1}$ are converging, diverging, monotone (increasing or decreasing) and/or bounded. Check all boxes that apply. convergent divergent monotone bounded (i) ${\frac {1}{n^{2}}}$ (ii) $(-1)^{n}+1$ (iii) $-1+2^{-n}$ (iv) $\ln(n)$

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Hint

If a sequence has a limit, we say the sequence is convergent, and that the sequence converges to the limit. Otherwise, the sequence is divergent. A monotone sequence: ${a_{n}}$ such that either (1) $a_{i+1}>=a_{i}$ for every $i\geq 1$, or (2) $a_{i+1}\leq a_{i}$ for every $i\geq 1$. A bounded sequence: A sequence ${x_{n}}$ is said to be bounded if $\exists M>0$ such that $|x_{n}|\leq M$ for all positive integers $n$

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. (i): when $n\to \infty$, sequence gets smaller, and goes to to 0. Thus monotone and convergent. ${x_{n}}>0$, the largest one is ${x_{1}}=1$, bounded. (ii) $(-1)^{n}={1,-1}$, so sequence will never converge to a limit (it will change switch between "2" and "0" ). But it do have a upper bound which is 2. (iii) $-1+(0.5)^{n}$. -1 is a constant, just leave it there. $0.5^{n}$ gets smaller with n increase, until converge to a limit 0. the range for $0.5^{n}$ is (0,0.5]. (iv) The graph of $ln(x)$ is monotone increasing. Thus it is not convergent and do not have a bound. $\color {blue}Answer:$ convergent divergent monotone bounded (i) ${\frac {1}{n^{2}}}$ Yes No Yes Yes (ii) $(-1)^{n}+1$ No Yes No Yes (iii) $-1+2^{-n}$ Yes No Yes Yes (iv) $\ln(n)$ No Yes Yes No