All my high school teachers
by djmathman, Jun 14, 2016, 8:51 PM
are either leaving or retiring. Why 
They were also considered to be some of the best teachers in the school, and especially ones who had a big influence on my high school years. I hope I can still communicate with them on a somewhat regular basis.
I'd also hate to see my old high school become a hollow shell of its former self....
They were also considered to be some of the best teachers in the school, and especially ones who had a big influence on my high school years. I hope I can still communicate with them on a somewhat regular basis.
I'd also hate to see my old high school become a hollow shell of its former self....
Woods Advanced Calculus Chapter 2 Problem 30 wrote:
Show that ![\[a_0+a_1+a_2+\cdots+a_n+\cdots\]](//latex.artofproblemsolving.com/2/a/2/2a2e9ae003faaca791f6579f8870e848812ff36e.png)
converges if ![$\lim_{n\to\infty}\sqrt[n]{a_n}<1$](//latex.artofproblemsolving.com/3/4/9/349d0f9d9adcfe5671bbd5f9be964f27b32fdbba.png)
and diverges if ![$\lim_{n\to\infty}\sqrt[n]{a_n}>1$](//latex.artofproblemsolving.com/9/8/a/98aca3e3293cd46e3c2519b5599a68e98b970bc5.png)
.
![\[a_0+a_1+a_2+\cdots+a_n+\cdots\]](http://latex.artofproblemsolving.com/2/a/2/2a2e9ae003faaca791f6579f8870e848812ff36e.png)
converges if ![$\lim_{n\to\infty}\sqrt[n]{a_n}<1$](http://latex.artofproblemsolving.com/3/4/9/349d0f9d9adcfe5671bbd5f9be964f27b32fdbba.png)
and diverges if ![$\lim_{n\to\infty}\sqrt[n]{a_n}>1$](http://latex.artofproblemsolving.com/9/8/a/98aca3e3293cd46e3c2519b5599a68e98b970bc5.png)
.
,![\[S_k = a_1+a_2+\cdots+a_k.\]](http://latex.artofproblemsolving.com/6/7/a/67a41d09fd7d4c43901ff5756e57597b3c4078fd.png)
Suppose ![$\lim_{n\to\infty}\sqrt[n]{a_n}=L$](http://latex.artofproblemsolving.com/e/e/4/ee4b145187d1bb42d85ccc2a238bb89090e3b6ae.png)
satisfies 
.
must be positive, since ![$\sqrt[n]{a_n}$](http://latex.artofproblemsolving.com/b/3/1/b31d89b8429a2da8628d4b2dcb1aa1c8fe4fbbc2.png)
must be positive for all 
even. By the definition of limit, for any 
,
such that for all 
we have ![\[|\sqrt[n]{a_n}-L| < \varepsilon\quad\iff\quad (L-\varepsilon)^n < a_n < (L + \varepsilon)^n.\]](http://latex.artofproblemsolving.com/d/e/8/de86b0d496431e0547d1635530d31f5bbe777c69.png)
Now, recall that by our definition of 
we have 
,
goes to zero as 
such that for all 
we have ![\[(L+\varepsilon)^n < \varepsilon(1-L-\varepsilon).\]](http://latex.artofproblemsolving.com/d/a/7/da7c1d32f9446c0e133c81993350c97e4b1b9f70.png)
This means that for all 
and 
,
Since 
is Cauchy - and hence convergent.
,
such that ![$\sqrt[n]{a_n} > 1$](http://latex.artofproblemsolving.com/b/0/5/b058ba4befc04c74a494daf37a232b7629d030fa.png)
for all 
.
.
But 
goes to infinity as 
,
)![\[\sum_{n=0}^\infty\dfrac{1}{n^2}\qquad\text{and}\qquad \sum_{n=0}^\infty\dfrac{1}{n}.\]](http://latex.artofproblemsolving.com/c/0/e/c0e781e13ba0c9db0a9fc15489cb153da1265378.png)
Both of these give 
is basically swallowed up by the force of the radical), but the first one converges while the second one diverges.
April 2025
November 2024