does this series converge?

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October 9th 2008, 10:28 PM
Cato

does this series converge?

$\sum {\frac {2+sin^3(n+1)} {2^n + n^2} }$

Can someone show me this plz?
October 9th 2008, 10:38 PM
Jhevon

Quote:

Originally Posted by Cato

$\sum {\frac {2+sin^3(n+1)} {2^n + n^2} }$

Can someone show me this plz?

note that $\Bigg| \frac {2 + \sin^3 (n + 1)}{2^n + n^2} \Bigg| \le \Bigg| \frac {2 + 1}{2^n + n^2} \Bigg| = \Bigg| \frac 3{2^n + n^2} \Bigg|$

now what can you say?
October 9th 2008, 10:57 PM
Cato

$\Bigg| \frac 3{2^n + n^2} \Bigg|$ < $\Bigg| \frac 3{n^2} \Bigg|$ which we know converges so the whole thing converges, right?
October 9th 2008, 11:40 PM
Jhevon

Quote:

Originally Posted by Cato

$\Bigg| \frac 3{2^n + n^2} \Bigg|$ < $\Bigg| \frac 3{n^2} \Bigg|$ which we know converges so the whole thing converges, right?

yes, it converges absolutely by the comparison test. and absolute convergence implies convergence

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