Finding the monotonicity of simple sequence - how to?
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I'm trying to find the monotonicity (whether it's increasing, decreasing or non-existeng) of such simple sequence: $$a_{n} = sqrt[n]{2^n+3^n}$$
$$frac{a_{n}}{a_{n+1}}=frac{sqrt[n]{2^n+3^n}}{sqrt[n+1]{2^{n+1}+3^{n+1}}}$$
I have dealt with such exercises before without problems. This one I have no idea how to proceed further.
Help is appreciated, thanks.
real-analysis sequences-and-series limits
asked Jan 1 at 23:42
JanPawelJanPawel
211
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add a comment |
$begingroup$
I'm trying to find the monotonicity (whether it's increasing, decreasing or non-existeng) of such simple sequence: $$a_{n} = sqrt[n]{2^n+3^n}$$
$$frac{a_{n}}{a_{n+1}}=frac{sqrt[n]{2^n+3^n}}{sqrt[n+1]{2^{n+1}+3^{n+1}}}$$
I have dealt with such exercises before without problems. This one I have no idea how to proceed further.
Help is appreciated, thanks.
real-analysis sequences-and-series limits
asked Jan 1 at 23:42
JanPawelJanPawel
211
$endgroup$
$begingroup$
Easily seen from $a_n=3sqrt[n]{(2/3)^n+1}$.
$endgroup$
– metamorphy
Jan 2 at 0:06
add a comment |
$begingroup$
I'm trying to find the monotonicity (whether it's increasing, decreasing or non-existeng) of such simple sequence: $$a_{n} = sqrt[n]{2^n+3^n}$$
$$frac{a_{n}}{a_{n+1}}=frac{sqrt[n]{2^n+3^n}}{sqrt[n+1]{2^{n+1}+3^{n+1}}}$$
I have dealt with such exercises before without problems. This one I have no idea how to proceed further.
Help is appreciated, thanks.
real-analysis sequences-and-series limits
asked Jan 1 at 23:42
JanPawelJanPawel
211
$endgroup$
I'm trying to find the monotonicity (whether it's increasing, decreasing or non-existeng) of such simple sequence: $$a_{n} = sqrt[n]{2^n+3^n}$$
$$frac{a_{n}}{a_{n+1}}=frac{sqrt[n]{2^n+3^n}}{sqrt[n+1]{2^{n+1}+3^{n+1}}}$$
I have dealt with such exercises before without problems. This one I have no idea how to proceed further.
Help is appreciated, thanks.
real-analysis sequences-and-series limits
real-analysis sequences-and-series limits
asked Jan 1 at 23:42
JanPawelJanPawel
211
asked Jan 1 at 23:42
JanPawelJanPawel
211
asked Jan 1 at 23:42
JanPawelJanPawel
211
asked Jan 1 at 23:42
JanPawelJanPawel
211
asked Jan 1 at 23:42
JanPawelJanPawel
211
211
$begingroup$
Easily seen from $a_n=3sqrt[n]{(2/3)^n+1}$.
$endgroup$
– metamorphy
Jan 2 at 0:06
add a comment |
$begingroup$
Easily seen from $a_n=3sqrt[n]{(2/3)^n+1}$.
$endgroup$
– metamorphy
Jan 2 at 0:06
$begingroup$
Easily seen from $a_n=3sqrt[n]{(2/3)^n+1}$.
$endgroup$
– metamorphy
Jan 2 at 0:06
$begingroup$
Easily seen from $a_n=3sqrt[n]{(2/3)^n+1}$.
$endgroup$
– metamorphy
Jan 2 at 0:06
add a comment |
1 Answer
1
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oldest
votes
$begingroup$
Rearrange things a bit
$$
a_n = 3left[left(frac{2}{3}right)^n + 1 right]^{1/n}
$$
As $n$ increases the term $(2/3)^n$ goes to zero and $a_n$ monotonically decreases to $3$
answered Jan 2 at 0:08
caveraccaverac
14.5k31130
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add a comment |
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1 Answer
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active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Rearrange things a bit
$$
a_n = 3left[left(frac{2}{3}right)^n + 1 right]^{1/n}
$$
As $n$ increases the term $(2/3)^n$ goes to zero and $a_n$ monotonically decreases to $3$
answered Jan 2 at 0:08
caveraccaverac
14.5k31130
$endgroup$
add a comment |
$begingroup$
Rearrange things a bit
$$
a_n = 3left[left(frac{2}{3}right)^n + 1 right]^{1/n}
$$
As $n$ increases the term $(2/3)^n$ goes to zero and $a_n$ monotonically decreases to $3$
answered Jan 2 at 0:08
caveraccaverac
14.5k31130
$endgroup$
add a comment |
$begingroup$
Rearrange things a bit
$$
a_n = 3left[left(frac{2}{3}right)^n + 1 right]^{1/n}
$$
As $n$ increases the term $(2/3)^n$ goes to zero and $a_n$ monotonically decreases to $3$
answered Jan 2 at 0:08
caveraccaverac
14.5k31130
$endgroup$
Rearrange things a bit
$$
a_n = 3left[left(frac{2}{3}right)^n + 1 right]^{1/n}
$$
As $n$ increases the term $(2/3)^n$ goes to zero and $a_n$ monotonically decreases to $3$
answered Jan 2 at 0:08
caveraccaverac
14.5k31130
answered Jan 2 at 0:08
caveraccaverac
14.5k31130
answered Jan 2 at 0:08
caveraccaverac
14.5k31130
answered Jan 2 at 0:08
caveraccaverac
14.5k31130
14.5k31130
add a comment |
add a comment |
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$begingroup$
Easily seen from $a_n=3sqrt[n]{(2/3)^n+1}$.
$endgroup$
– metamorphy
Jan 2 at 0:06