Determine whether the sequence converges or diverges.
Clash Royale CLAN TAG#URR8PPP
$begingroup$
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:
calculus sequences-and-series
edited Jan 15 at 5:06
user587192
1,978315
asked Jan 15 at 3:42
airlanggaairlangga
434
$endgroup$
add a comment |
$begingroup$
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:
calculus sequences-and-series
edited Jan 15 at 5:06
user587192
1,978315
asked Jan 15 at 3:42
airlanggaairlangga
434
$endgroup$
$begingroup$
@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
add a comment |
$begingroup$
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:
calculus sequences-and-series
edited Jan 15 at 5:06
user587192
1,978315
asked Jan 15 at 3:42
airlanggaairlangga
434
$endgroup$
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:
calculus sequences-and-series
calculus sequences-and-series
edited Jan 15 at 5:06
user587192
1,978315
asked Jan 15 at 3:42
airlanggaairlangga
434
edited Jan 15 at 5:06
user587192
1,978315
asked Jan 15 at 3:42
airlanggaairlangga
434
edited Jan 15 at 5:06
user587192
1,978315
edited Jan 15 at 5:06
user587192
1,978315
edited Jan 15 at 5:06
user587192
1,978315
1,978315
asked Jan 15 at 3:42
airlanggaairlangga
434
asked Jan 15 at 3:42
airlanggaairlangga
434
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
add a comment |
$begingroup$
@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
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
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
$$ left| frac (-1)^n pi^n 4^n right| = frac pi^n 4^n = left( frac pi 4 right)^n to 0 $$
answered Jan 15 at 3:48
Jimmy SabaterJimmy Sabater
2,640321
$endgroup$
add a comment |
$begingroup$
Well: $$|t|<1to lim_ntoinftyt^n=0$$
answered Jan 15 at 4:04
Rhys HughesRhys Hughes
5,7481529
$endgroup$
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
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active
oldest
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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
$$ left| frac (-1)^n pi^n 4^n right| = frac pi^n 4^n = left( frac pi 4 right)^n to 0 $$
answered Jan 15 at 3:48
Jimmy SabaterJimmy Sabater
2,640321
$endgroup$
add a comment |
$begingroup$
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
$$ left| frac (-1)^n pi^n 4^n right| = frac pi^n 4^n = left( frac pi 4 right)^n to 0 $$
answered Jan 15 at 3:48
Jimmy SabaterJimmy Sabater
2,640321
$endgroup$
add a comment |
$begingroup$
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
$$ left| frac (-1)^n pi^n 4^n right| = frac pi^n 4^n = left( frac pi 4 right)^n to 0 $$
answered Jan 15 at 3:48
Jimmy SabaterJimmy Sabater
2,640321
$endgroup$
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
$$ left| frac (-1)^n pi^n 4^n right| = frac pi^n 4^n = left( frac pi 4 right)^n to 0 $$
answered Jan 15 at 3:48
Jimmy SabaterJimmy Sabater
2,640321
answered Jan 15 at 3:48
Jimmy SabaterJimmy Sabater
2,640321
answered Jan 15 at 3:48
Jimmy SabaterJimmy Sabater
2,640321
answered Jan 15 at 3:48
Jimmy SabaterJimmy Sabater
2,640321
2,640321
add a comment |
add a comment |
$begingroup$
Well: $$|t|<1to lim_ntoinftyt^n=0$$
answered Jan 15 at 4:04
Rhys HughesRhys Hughes
5,7481529
$endgroup$
add a comment |
$begingroup$
Well: $$|t|<1to lim_ntoinftyt^n=0$$
answered Jan 15 at 4:04
Rhys HughesRhys Hughes
5,7481529
$endgroup$
add a comment |
$begingroup$
Well: $$|t|<1to lim_ntoinftyt^n=0$$
answered Jan 15 at 4:04
Rhys HughesRhys Hughes
5,7481529
$endgroup$
Well: $$|t|<1to lim_ntoinftyt^n=0$$
answered Jan 15 at 4:04
Rhys HughesRhys Hughes
5,7481529
answered Jan 15 at 4:04
Rhys HughesRhys Hughes
5,7481529
answered Jan 15 at 4:04
Rhys HughesRhys Hughes
5,7481529
answered Jan 15 at 4:04
Rhys HughesRhys Hughes
5,7481529
5,7481529
add a comment |
add a comment |
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$begingroup$
@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