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Let ${x_n}$ be the sequence $+\sqrt{1}, -\sqrt{1},+\sqrt{2}, -\sqrt{2},+\sqrt{3}, -\sqrt{3}$ ...

If

$$y_n = \frac{{x_1}+{x_2}+...+{x_n}}{n}$$ for all $n \in\Bbb N$, then the sequence $\{y_n\}$ is:

a) Monotonic or b) Not Bounded or c) Bounded but not Convergent or d) Convergent.

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    Calculate $y_1$, $y_2$, $y_3$, $y_4$, $y_5$, $y_6$. (No calculaor, just observe the obvious cancellations.) I think the answers will become clear.2012-08-14
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    This question was in my exam. Now,I can try to solve it ...thanks2012-08-14
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    Well clearly a) is not the answer, since if the sequence is monotonic then it is either not bounded or convergent.2012-08-14

1 Answers 1

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If you don’t immediately see what’s going on, always try calculating $y_n$ for small $n$:

$$\begin{align*} y_1&=\sqrt 1=1\\ y_2&=0\\ y_3&=\frac{\sqrt2}3\\ y_4&=0\\ y_5&=\frac{\sqrt3}5\\ y_6&=0\\ y_7&=\frac{\sqrt4}7 \end{align*}$$

Look for patterns, and then try to see why they’re there. Here you shouldn’t have too much trouble spotting the patterns and writing down a general formula for $y_n$ when $n$ is even and another for $y_n$ when $n$ is odd. Even with the seven values that I’ve calculated here you should be able to explain why (a) is not the answer, and once you figure out the general formulas, the right answer is easy to pick out.

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    I think, the ansewr must be (d), because $y_n:=\begin{cases} 0 & if\,n\,is\,even.\\ \frac{\sqrt {(n+1)/2}}{ n} & if\, n \,is\, odd.\\ \end{cases}$ $\implies\lim_{\,\,n\to\infty} y_n = 0.$ $\implies y_n$ converges to $0$.2012-08-14
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    @ram: Yes, that does it just it just fine. (Sorry that I didn’t notice that you had the indexing wrong the first time.)2012-08-14
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    ok...Sir. Thanks a lot.2012-08-14