- #1
ognik
- 643
- 2
I've just studied integral tests for convergence, 1st timer, but some detail is escaping me.
The text reads:
1. Show that if $ \lim_{{n}\to{\infty}} {n}^{p}\: {U}_{n}\implies A \lt \infty\: (p \gt 1) $
Then $ \sum_{n=1}^{\infty} {U}_{n}\: $ converges
2. Show that if $ \lim_{{n}\to{\infty}} n {U}_{n}\implies A \gt 0 $ the series diverges
"These two tests, known as limit tests, are often convenient for establishing
the convergence of a series. They may be treated as comparison
tests"
--------------------
I can 'see' these are true, but formal proof escapes me. I tried 'limit of products=product of limits':
$ \lim_{{n}\to{\infty}}{n}^{p}{U}_{n} = \lim_{{n}\to{\infty}} {n}^{p}\lim_{{n}\to{\infty}}{U}_{n} \implies A $
$ \therefore \lim_{{n}\to{\infty}}{U}_{n} \implies A/ \lim_{{n}\to{\infty}} {n}^{p} $
Clearly $ \lim_{{n}\to{\infty}} {n}^{p} \implies\infty\: for\: p \gt 1 $
$ \therefore \lim_{{n}\to{\infty}}{U}_{n} \implies A/\infty = 0 $
But I'm stuck here, how do I relate the above result to $ \sum_{n=1}^{\infty} {U}_{n}\: $ ?
Appreciate the help, thanks.
The text reads:
1. Show that if $ \lim_{{n}\to{\infty}} {n}^{p}\: {U}_{n}\implies A \lt \infty\: (p \gt 1) $
Then $ \sum_{n=1}^{\infty} {U}_{n}\: $ converges
2. Show that if $ \lim_{{n}\to{\infty}} n {U}_{n}\implies A \gt 0 $ the series diverges
"These two tests, known as limit tests, are often convenient for establishing
the convergence of a series. They may be treated as comparison
tests"
--------------------
I can 'see' these are true, but formal proof escapes me. I tried 'limit of products=product of limits':
$ \lim_{{n}\to{\infty}}{n}^{p}{U}_{n} = \lim_{{n}\to{\infty}} {n}^{p}\lim_{{n}\to{\infty}}{U}_{n} \implies A $
$ \therefore \lim_{{n}\to{\infty}}{U}_{n} \implies A/ \lim_{{n}\to{\infty}} {n}^{p} $
Clearly $ \lim_{{n}\to{\infty}} {n}^{p} \implies\infty\: for\: p \gt 1 $
$ \therefore \lim_{{n}\to{\infty}}{U}_{n} \implies A/\infty = 0 $
But I'm stuck here, how do I relate the above result to $ \sum_{n=1}^{\infty} {U}_{n}\: $ ?
Appreciate the help, thanks.