In order to prove that the sequence $a_n$=($1^k+2^k+3^k+..n^k$)$/n!$ is bounded(k is fixed natural number).Is it enough to say that $a_n
Boundedness of sequences
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0$M$ can depend on $k$, but not on $n$. Otherwise, you could just say something like $|a_n|\lt1+(1^k+\cdots+n^k)/n!$. – 2012-10-25
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0Ok, I was thinking the same thing,but there were misleading opinions. Then what do you think is the bound for this sequence? – 2012-10-25
2 Answers
$M$ should be a constant that does not depend on $n$. It's true that $a_n \le n^{k+1}/n!$, but that doesn't answer the question yet. If you can show that the sequence $\{n^{k+1}/n!\}_{n \ge 1}$ is bounded by some $M$ (that does not depend on $n$), then we have $$0 \le a_n \le \frac{n^{k+1}}{n!} \le M,$$ which does prove that $\{a_n\}$ is bounded.
The more general fact that you are using is:
If $\{a_n\}$ and $\{b_n\}$ are nonnegative sequences with $\{b_n\}$ bounded and $a_n \le b_n$ for all $n$, then $\{a_n\}$ is also bounded.
You are correct to say that $|a_n| < n^{k+1} / n!$, but this does not show the function is bounded by some constant (i.e. not depending on $n$). To find such a bound, we can use the fact that there is $n_0$ such that $n^{k+1} / n! < 1$ for all $n \geq n_0$ (i.e. $n!$ eventually dominates $n^{k+1}$ for constant $k$). We can therefore let $$ A = \bigcup_{i = 1}^{n_0} \{i^{k+1} / i!\} \cup \{1\} $$ and take $$ M = \max A. $$ That is, we choose as our bound one of the first $n_0$ terms of the sequence or $1$, whichever is largest.
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0That is a nice way of showing the bound.Thank you Austin :) – 2012-10-25
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0But what if k=n or k>n,then $n!$ is always smaller than $n^{k+1}$ – 2012-10-25
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1@Pilot: $k$ is a constant. $n$ keeps growing, but $k$ never changes. – 2012-10-25
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0What if $k=n$? WIll it be still bounded? – 2012-10-25
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0Can I say,it is divergent because $a_n>n^n/n!$ and$n^n/n!$ diverges,thats why $a_n$ diverges? – 2012-10-25
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0@Pilot In that case, you would be correct. That's why it is very important that $k$ is a fixed constant here. – 2012-10-25