Determine all values of $w$ for which $sumlimits_{n=1}^{infty}left(frac2nright)^w$ converges.
$begingroup$
I need some help for these following connected questions in my calc workbook.
The answer format is supposed to be in interval notation.
1) Determine all values of $w$ for which$displaystylesum_{n=1}^{infty}left(dfrac{2}{w}right)^n$ converges.
I know the lower bound for w will have to be 2 for the fraction to be possible and i thought the upper bound would be inf but it was incorrect. I don't know where to go off on this?
2) Determine all values of $w$ for which $displaystylesum_{n=1}^{infty}left(dfrac{2}{n}right)^w$ converges.
I also don't know where to start on this. Would w have to be negative numbers?
Thank you in advance!
calculus sequences-and-series convergence
edited Jan 20 at 9:04
Martin Sleziak
44.7k10118272
asked Jan 20 at 1:19
ninjagirlninjagirl
155110
$endgroup$
add a comment |
$begingroup$
I need some help for these following connected questions in my calc workbook.
The answer format is supposed to be in interval notation.
1) Determine all values of $w$ for which$displaystylesum_{n=1}^{infty}left(dfrac{2}{w}right)^n$ converges.
I know the lower bound for w will have to be 2 for the fraction to be possible and i thought the upper bound would be inf but it was incorrect. I don't know where to go off on this?
2) Determine all values of $w$ for which $displaystylesum_{n=1}^{infty}left(dfrac{2}{n}right)^w$ converges.
I also don't know where to start on this. Would w have to be negative numbers?
Thank you in advance!
calculus sequences-and-series convergence
edited Jan 20 at 9:04
Martin Sleziak
44.7k10118272
asked Jan 20 at 1:19
ninjagirlninjagirl
155110
$endgroup$
add a comment |
$begingroup$
I need some help for these following connected questions in my calc workbook.
The answer format is supposed to be in interval notation.
1) Determine all values of $w$ for which$displaystylesum_{n=1}^{infty}left(dfrac{2}{w}right)^n$ converges.
I know the lower bound for w will have to be 2 for the fraction to be possible and i thought the upper bound would be inf but it was incorrect. I don't know where to go off on this?
2) Determine all values of $w$ for which $displaystylesum_{n=1}^{infty}left(dfrac{2}{n}right)^w$ converges.
I also don't know where to start on this. Would w have to be negative numbers?
Thank you in advance!
calculus sequences-and-series convergence
edited Jan 20 at 9:04
Martin Sleziak
44.7k10118272
asked Jan 20 at 1:19
ninjagirlninjagirl
155110
$endgroup$
I need some help for these following connected questions in my calc workbook.
The answer format is supposed to be in interval notation.
1) Determine all values of $w$ for which$displaystylesum_{n=1}^{infty}left(dfrac{2}{w}right)^n$ converges.
I know the lower bound for w will have to be 2 for the fraction to be possible and i thought the upper bound would be inf but it was incorrect. I don't know where to go off on this?
2) Determine all values of $w$ for which $displaystylesum_{n=1}^{infty}left(dfrac{2}{n}right)^w$ converges.
I also don't know where to start on this. Would w have to be negative numbers?
Thank you in advance!
calculus sequences-and-series convergence
calculus sequences-and-series convergence
edited Jan 20 at 9:04
Martin Sleziak
44.7k10118272
asked Jan 20 at 1:19
ninjagirlninjagirl
155110
edited Jan 20 at 9:04
Martin Sleziak
44.7k10118272
asked Jan 20 at 1:19
ninjagirlninjagirl
155110
edited Jan 20 at 9:04
Martin Sleziak
44.7k10118272
edited Jan 20 at 9:04
Martin Sleziak
44.7k10118272
edited Jan 20 at 9:04
Martin Sleziak
44.7k10118272
44.7k10118272
asked Jan 20 at 1:19
ninjagirlninjagirl
155110
asked Jan 20 at 1:19
ninjagirlninjagirl
155110
asked Jan 20 at 1:19
ninjagirlninjagirl
155110
155110
add a comment |
add a comment |
1 Answer
1
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$begingroup$
Use the root test for 1). We have
$$ lim sqrt[n]{a_n} = frac{2}{w} $$
Thus, as long as $w > 2$, we have convergence. For 2), notice that the sum is equivalent to
$$ 2^w sum frac{1}{n^w} $$
which converges as long as $w>1$ (p-series)
answered Jan 20 at 1:25
Jimmy SabaterJimmy Sabater
2,645323
$endgroup$
$begingroup$
for one, that's what I thought the answer was and put in (2, inf) but my answer was deemed incorrect. any ideas to why it might not be accepting my answer?
$endgroup$
– ninjagirl
Jan 20 at 1:40
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Use the root test for 1). We have
$$ lim sqrt[n]{a_n} = frac{2}{w} $$
Thus, as long as $w > 2$, we have convergence. For 2), notice that the sum is equivalent to
$$ 2^w sum frac{1}{n^w} $$
which converges as long as $w>1$ (p-series)
answered Jan 20 at 1:25
Jimmy SabaterJimmy Sabater
2,645323
$endgroup$
$begingroup$
for one, that's what I thought the answer was and put in (2, inf) but my answer was deemed incorrect. any ideas to why it might not be accepting my answer?
$endgroup$
– ninjagirl
Jan 20 at 1:40
add a comment |
$begingroup$
Use the root test for 1). We have
$$ lim sqrt[n]{a_n} = frac{2}{w} $$
Thus, as long as $w > 2$, we have convergence. For 2), notice that the sum is equivalent to
$$ 2^w sum frac{1}{n^w} $$
which converges as long as $w>1$ (p-series)
answered Jan 20 at 1:25
Jimmy SabaterJimmy Sabater
2,645323
$endgroup$
$begingroup$
for one, that's what I thought the answer was and put in (2, inf) but my answer was deemed incorrect. any ideas to why it might not be accepting my answer?
$endgroup$
– ninjagirl
Jan 20 at 1:40
add a comment |
$begingroup$
Use the root test for 1). We have
$$ lim sqrt[n]{a_n} = frac{2}{w} $$
Thus, as long as $w > 2$, we have convergence. For 2), notice that the sum is equivalent to
$$ 2^w sum frac{1}{n^w} $$
which converges as long as $w>1$ (p-series)
answered Jan 20 at 1:25
Jimmy SabaterJimmy Sabater
2,645323
$endgroup$
Use the root test for 1). We have
$$ lim sqrt[n]{a_n} = frac{2}{w} $$
Thus, as long as $w > 2$, we have convergence. For 2), notice that the sum is equivalent to
$$ 2^w sum frac{1}{n^w} $$
which converges as long as $w>1$ (p-series)
answered Jan 20 at 1:25
Jimmy SabaterJimmy Sabater
2,645323
answered Jan 20 at 1:25
Jimmy SabaterJimmy Sabater
2,645323
answered Jan 20 at 1:25
Jimmy SabaterJimmy Sabater
2,645323
answered Jan 20 at 1:25
Jimmy SabaterJimmy Sabater
2,645323
2,645323
$begingroup$
for one, that's what I thought the answer was and put in (2, inf) but my answer was deemed incorrect. any ideas to why it might not be accepting my answer?
$endgroup$
– ninjagirl
Jan 20 at 1:40
add a comment |
$begingroup$
for one, that's what I thought the answer was and put in (2, inf) but my answer was deemed incorrect. any ideas to why it might not be accepting my answer?
$endgroup$
– ninjagirl
Jan 20 at 1:40
$begingroup$
for one, that's what I thought the answer was and put in (2, inf) but my answer was deemed incorrect. any ideas to why it might not be accepting my answer?
$endgroup$
– ninjagirl
Jan 20 at 1:40
$begingroup$
for one, that's what I thought the answer was and put in (2, inf) but my answer was deemed incorrect. any ideas to why it might not be accepting my answer?
$endgroup$
– ninjagirl
Jan 20 at 1:40
add a comment |
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