converges,diverges....... help!

[chuang tsai-ling]

1. Prove that for an eventually decreasing sequence{a_n}:
{a_n} is bounded below by M,in which case there exists L≧M such that lim(n→∞)a<sub>n=L

2. Consider the sequence {</sub>a_{n},where a weighted average define} a_{n+2 =} 〔2 ₍a_{n+1) +(}a_n)〕 _{/ 3
with n} ∈N and ∈a_{1 and} a_{2 any real numbers.Prove that {}a_{n} converges to (1/4)}a_1+(3/4)a<sub>2

3.Label each statement as true or false. If a statement is true ,prove it .If not ,
(1)give an example of why it is false,and
(2)if possible, correct it to make it true, and then prove it.

Q1: The sequence {</sub>a_{n} with} a_n=(-1)ⁿ/n is an oscillating sequence.

Q2 :If {a_n} and {b_n} both diverge,then {a_nb_n} diverges.


Thanks for helping me
question 1 and 2,I have no idea how to start them.
and question 3 like a definition ,how to prove ><

Thanks！

 

[HallsofIvy]

1) if the sequence \(\displaystyle \{a_n\}\) has lower bound, M, then it has a greatest lower bound. Call that greatest lower bound, L.

Given any \(\displaystyle \epsilon> 0\) there exist N such that \(\displaystyle L+\epsilon\ge a_N> L\). Why? What would be true if there were no such N?

2) Given that \(\displaystyle a_{n+ 2}= \frac{2}{3}a_{n+1}+ \frac{1}{3}a_n\), suppose that \(\displaystyle \{a_n\}\) converges to A. What would
\(\displaystyle \lim_{n\to\infty} a_{n+2}= \frac{2}{3} \lim_{n\to\infty} + \frac{1}{3} \lim_{n\to\infty} a_{n+ 1}\)?

Use that information to show that it does converge.

3) Q1 What is the definition of "alternating sequence"?
Q2 Look at \(\displaystyle a_n= (-1)^n\) and \(\displaystyle b_n= (-1)^{n+1}\)

 

[chuang tsai-ling]

1) if the sequence \(\displaystyle \{a_n\}\) has lower bound, M, then it has a greatest lower bound. Call that greatest lower bound, L.

Given any \(\displaystyle \epsilon> 0\) there exist N such that \(\displaystyle L+\epsilon\ge a_N> L\). Why? What would be true if there were no such N?

2) Given that \(\displaystyle a_{n+ 2}= \frac{2}{3}a_{n+1}+ \frac{1}{3}a_n\), suppose that \(\displaystyle \{a_n\}\) converges to A. What would
\(\displaystyle \lim_{n\to\infty} a_{n+2}= \frac{2}{3} \lim_{n\to\infty} + \frac{1}{3} \lim_{n\to\infty} a_{n+ 1}\)?

Use that information to show that it does converge.

3) Q1 What is the definition of "alternating sequence"?
Q2 Look at \(\displaystyle a_n= (-1)^n\) and \(\displaystyle b_n= (-1)^{n+1}\)



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and i don't know the meaning of a_{1 and a2 . thanks}


[FONT=MathJax_Main]lim[/FONT][FONT=MathJax_Math-italic]n[/FONT][FONT=MathJax_Main]→[/FONT][FONT=MathJax_Main]∞[/FONT][FONT=MathJax_Math-italic]a[/FONT][FONT=MathJax_Math-italic]n[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main]lim[/FONT][FONT=MathJax_Math-italic]n[/FONT][FONT=MathJax_Main]→[/FONT][FONT=MathJax_Main]∞[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main]lim[/FONT][FONT=MathJax_Math-italic]n[/FONT][FONT=MathJax_Main]→[/FONT][FONT=MathJax_Main]∞[/FONT][FONT=MathJax_Math-italic]a[/FONT][FONT=MathJax_Math-italic]n[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]1[/FONT]?