A question on sequences

• March 14th 2009, 09:59 PM
twilightstr
$a_n= (3^n + 5^n)^{\frac 1n}$

is the above sequence convergent? if so, what is the limit?
• March 14th 2009, 10:41 PM
Jhevon
Quote:

Originally Posted by twilightstr
$a_n= (3^n + 5^n)^{\frac 1n}$

is the above sequence convergent? if so, what is the limit?

it is convergent.

Hint: $(3^n + 5^n)^{\frac 1n} = \text{exp} \left( \ln (3^n + 5^n)^{\frac 1n} \right) = \text{exp} \left( \frac {\ln (3^n + 5^n)}n \right)$

now take the limit
• March 14th 2009, 11:10 PM
twilightstr
is it infinity
• March 14th 2009, 11:12 PM
Jhevon
Quote:

Originally Posted by twilightstr
is it infinity

no (if it were, it wouldn't be convergent)

Hint 2: you can use L'Hopital's rule to find the limit
• March 14th 2009, 11:18 PM
o_O
Just as an alternative..

If you can use the fact that $\lim_{n \to \infty} a^{\frac{1}{n}} = 1$ for $a > 0$, then notice that:

$5^n \ \ \leq \ \ 3^n + 5^n \ \ \leq \ \ 5^n + 5^n$

$\Rightarrow \left(5^n \right)^{\frac{1}{n}} \ \ \leq \ \ \left(3^n +5^n \right)^{\frac{1}{n}} \ \ \leq \ \ \left(5^n +5^n \right)^{\frac{1}{n}}$

$\Rightarrow 5 \ \ \leq \ \ \left(3^n +5^n \right)^{\frac{1}{n}} \ \ \leq \ \ 5\left(2 \right)^{\frac{1}{n}}$

And use squeeze theorem ..
• March 14th 2009, 11:23 PM
twilightstr
jhevon, how would u take the limit
• March 14th 2009, 11:34 PM
Jhevon
Quote:

Originally Posted by twilightstr
jhevon, how would u take the limit

I told you. I would use L'Hopital's rule. o_O gives a nice alternative
• March 14th 2009, 11:36 PM
twilightstr
yes. thats what i tried doing in the first place, but i think i did it incorrectly.
• March 15th 2009, 12:23 AM
Jhevon
Quote:

Originally Posted by twilightstr
yes. thats what i tried doing in the first place, but i think i did it incorrectly.

recall that, by L'Hopital's, $\lim_{n \to \infty} \frac {\ln (3^n + 5^n)}n = \lim_{n \to \infty} \frac {\frac d{dn}[ \ln (3^n + 5^n)]}{\frac d{dn}n}$

i suppose it is the $\frac d{dn}[ \ln (3^n + 5^n)]$ that is giving you trouble. what did you get for this?
• March 15th 2009, 10:19 AM
stapel
Quote:

Originally Posted by twilightstr
yes. thats what i tried doing in the first place, but i think i did it incorrectly.

We'll be glad to look for any errors, but you'll need to show the work you did.

Please be complete. Thank you! :D
• March 15th 2009, 12:50 PM
twilightstr
= [(ln3)3^x + (ln5)5^x]/(3^x + 5^x) as the limit goes to infinity. I really dont know what to do from there
• March 15th 2009, 12:53 PM
Jhevon
Quote:

Originally Posted by twilightstr
= [(ln3)3^x + (ln5)5^x]/(3^x + 5^x) as the limit goes to infinity. I really dont know what to do from there

well, after that, we're good. multiply by $\frac {\frac 1{5^x}}{\frac 1{5^x}}$, the limit should seem obvious from there
• March 15th 2009, 01:05 PM
Krizalid
$\left( 3^{n}+5^{n} \right)^{1/n}=5\left\{ \left( \frac{3}{5} \right)^{n}+1 \right\}^{1/n}\to 5$ since $\left( \frac{3}{5} \right)^{n}+1\to 1$ and $1^{1/n}\to 1$ as $n\to\infty.$