Determine whether the sequence converges or diverges.

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP












2












$begingroup$


Let : $\a_n=frac(-pi)^n4^n$



What my teacher told me :



$a_1=frac-pi4 approx -0.785\
a_2=fracpi^216 approx 0.617\
a_3=frac-pi^364 approx -0.484\
a_4=fracpi^4256 approx 0.380\$



So the sequence diverges. But I'm not really sure about the answer.



Here is my teacher's work:



Here's my teacher's work






calculus sequences-and-series







share|cite|improve this question











$endgroup$











  • $begingroup$
    @Clclstdnt: this is a sequence, not a series. No alternating series test.
    $endgroup$
    – user587192
    Jan 15 at 3:50







  • 1




    $begingroup$
    One would not simply list four terms and claim the sequence is divergent or convergent. You may have probably taken a wrong note on what your teacher said.
    $endgroup$
    – user587192
    Jan 15 at 3:52











  • $begingroup$
    @user587192 I just attached her work
    $endgroup$
    – airlangga
    Jan 15 at 4:22










  • $begingroup$
    @airlangga: Your teacher's point is that $fracpi4<1$. And the sequence converges.
    $endgroup$
    – user587192
    Jan 15 at 4:26
















2












$begingroup$


Let : $\a_n=frac(-pi)^n4^n$



What my teacher told me :



$a_1=frac-pi4 approx -0.785\
a_2=fracpi^216 approx 0.617\
a_3=frac-pi^364 approx -0.484\
a_4=fracpi^4256 approx 0.380\$



So the sequence diverges. But I'm not really sure about the answer.



Here is my teacher's work:



Here's my teacher's work






calculus sequences-and-series







share|cite|improve this question











$endgroup$











  • $begingroup$
    @Clclstdnt: this is a sequence, not a series. No alternating series test.
    $endgroup$
    – user587192
    Jan 15 at 3:50







  • 1




    $begingroup$
    One would not simply list four terms and claim the sequence is divergent or convergent. You may have probably taken a wrong note on what your teacher said.
    $endgroup$
    – user587192
    Jan 15 at 3:52











  • $begingroup$
    @user587192 I just attached her work
    $endgroup$
    – airlangga
    Jan 15 at 4:22










  • $begingroup$
    @airlangga: Your teacher's point is that $fracpi4<1$. And the sequence converges.
    $endgroup$
    – user587192
    Jan 15 at 4:26














2












2








2





$begingroup$


Let : $\a_n=frac(-pi)^n4^n$



What my teacher told me :



$a_1=frac-pi4 approx -0.785\
a_2=fracpi^216 approx 0.617\
a_3=frac-pi^364 approx -0.484\
a_4=fracpi^4256 approx 0.380\$



So the sequence diverges. But I'm not really sure about the answer.



Here is my teacher's work:



Here's my teacher's work






calculus sequences-and-series







share|cite|improve this question











$endgroup$




Let : $\a_n=frac(-pi)^n4^n$



What my teacher told me :



$a_1=frac-pi4 approx -0.785\
a_2=fracpi^216 approx 0.617\
a_3=frac-pi^364 approx -0.484\
a_4=fracpi^4256 approx 0.380\$



So the sequence diverges. But I'm not really sure about the answer.



Here is my teacher's work:



Here's my teacher's work






calculus sequences-and-series




calculus sequences-and-series






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 15 at 5:06









user587192

1,978315




1,978315










asked Jan 15 at 3:42









airlanggaairlangga

434




434











  • $begingroup$
    @Clclstdnt: this is a sequence, not a series. No alternating series test.
    $endgroup$
    – user587192
    Jan 15 at 3:50







  • 1




    $begingroup$
    One would not simply list four terms and claim the sequence is divergent or convergent. You may have probably taken a wrong note on what your teacher said.
    $endgroup$
    – user587192
    Jan 15 at 3:52











  • $begingroup$
    @user587192 I just attached her work
    $endgroup$
    – airlangga
    Jan 15 at 4:22










  • $begingroup$
    @airlangga: Your teacher's point is that $fracpi4<1$. And the sequence converges.
    $endgroup$
    – user587192
    Jan 15 at 4:26

















  • $begingroup$
    @Clclstdnt: this is a sequence, not a series. No alternating series test.
    $endgroup$
    – user587192
    Jan 15 at 3:50







  • 1




    $begingroup$
    One would not simply list four terms and claim the sequence is divergent or convergent. You may have probably taken a wrong note on what your teacher said.
    $endgroup$
    – user587192
    Jan 15 at 3:52











  • $begingroup$
    @user587192 I just attached her work
    $endgroup$
    – airlangga
    Jan 15 at 4:22










  • $begingroup$
    @airlangga: Your teacher's point is that $fracpi4<1$. And the sequence converges.
    $endgroup$
    – user587192
    Jan 15 at 4:26
















$begingroup$
@Clclstdnt: this is a sequence, not a series. No alternating series test.
$endgroup$
– user587192
Jan 15 at 3:50





$begingroup$
@Clclstdnt: this is a sequence, not a series. No alternating series test.
$endgroup$
– user587192
Jan 15 at 3:50





1




1




$begingroup$
One would not simply list four terms and claim the sequence is divergent or convergent. You may have probably taken a wrong note on what your teacher said.
$endgroup$
– user587192
Jan 15 at 3:52





$begingroup$
One would not simply list four terms and claim the sequence is divergent or convergent. You may have probably taken a wrong note on what your teacher said.
$endgroup$
– user587192
Jan 15 at 3:52













$begingroup$
@user587192 I just attached her work
$endgroup$
– airlangga
Jan 15 at 4:22




$begingroup$
@user587192 I just attached her work
$endgroup$
– airlangga
Jan 15 at 4:22












$begingroup$
@airlangga: Your teacher's point is that $fracpi4<1$. And the sequence converges.
$endgroup$
– user587192
Jan 15 at 4:26





$begingroup$
@airlangga: Your teacher's point is that $fracpi4<1$. And the sequence converges.
$endgroup$
– user587192
Jan 15 at 4:26











2 Answers
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$begingroup$

By triangle inequality one has that if $lim_n to infty |a_n| = 0$, then $lim_n to infty a_n = 0$. Thus, if $a_n = frac(- pi )^n 4^n = frac (-1)^n pi^n 4^n $. Observe that



$$ left| frac (-1)^n pi^n 4^n right| = frac pi^n 4^n = left( frac pi 4 right)^n to 0 $$






share|cite|improve this answer









$endgroup$




















    3












    $begingroup$

    Well: $$|t|<1to lim_ntoinftyt^n=0$$






    share|cite|improve this answer









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      2 Answers
      2






      active

      oldest

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      2 Answers
      2






      active

      oldest

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      active

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      active

      oldest

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      3












      $begingroup$

      By triangle inequality one has that if $lim_n to infty |a_n| = 0$, then $lim_n to infty a_n = 0$. Thus, if $a_n = frac(- pi )^n 4^n = frac (-1)^n pi^n 4^n $. Observe that



      $$ left| frac (-1)^n pi^n 4^n right| = frac pi^n 4^n = left( frac pi 4 right)^n to 0 $$






      share|cite|improve this answer









      $endgroup$

















        3












        $begingroup$

        By triangle inequality one has that if $lim_n to infty |a_n| = 0$, then $lim_n to infty a_n = 0$. Thus, if $a_n = frac(- pi )^n 4^n = frac (-1)^n pi^n 4^n $. Observe that



        $$ left| frac (-1)^n pi^n 4^n right| = frac pi^n 4^n = left( frac pi 4 right)^n to 0 $$






        share|cite|improve this answer









        $endgroup$















          3












          3








          3





          $begingroup$

          By triangle inequality one has that if $lim_n to infty |a_n| = 0$, then $lim_n to infty a_n = 0$. Thus, if $a_n = frac(- pi )^n 4^n = frac (-1)^n pi^n 4^n $. Observe that



          $$ left| frac (-1)^n pi^n 4^n right| = frac pi^n 4^n = left( frac pi 4 right)^n to 0 $$






          share|cite|improve this answer









          $endgroup$



          By triangle inequality one has that if $lim_n to infty |a_n| = 0$, then $lim_n to infty a_n = 0$. Thus, if $a_n = frac(- pi )^n 4^n = frac (-1)^n pi^n 4^n $. Observe that



          $$ left| frac (-1)^n pi^n 4^n right| = frac pi^n 4^n = left( frac pi 4 right)^n to 0 $$







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Jan 15 at 3:48









          Jimmy SabaterJimmy Sabater

          2,640321




          2,640321





















              3












              $begingroup$

              Well: $$|t|<1to lim_ntoinftyt^n=0$$






              share|cite|improve this answer









              $endgroup$

















                3












                $begingroup$

                Well: $$|t|<1to lim_ntoinftyt^n=0$$






                share|cite|improve this answer









                $endgroup$















                  3












                  3








                  3





                  $begingroup$

                  Well: $$|t|<1to lim_ntoinftyt^n=0$$






                  share|cite|improve this answer









                  $endgroup$



                  Well: $$|t|<1to lim_ntoinftyt^n=0$$







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Jan 15 at 4:04









                  Rhys HughesRhys Hughes

                  5,7481529




                  5,7481529



























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