Prove $(a_n,b_n >0) land sum a_n $ converges $ land sum b_n $ diverges$implies liminflimits_{nrightarrow...
$(a_n)_{n in mathbb{N}},(b_n)_{n in mathbb{N}}$ are positive, real sequences!
Since $sum a_n$ converges, we know $limlimits_{nrightarrow infty}a_n =0$.
If $limlimits_{nrightarrow infty}b_n ne 0$ then $ liminflimits_{nrightarrow infty} frac{a_n}{b_n}=0$ is rather easy to show.
So let's assume $b_n$ converges to $0$. Since $sum a_n$ converges but $sum b_n$ does not, $a_n$ must somehow "converge faster" to $0$ than $b_n$ does, thus causing $ liminflimits_{nrightarrow infty} frac{a_n}{b_n}=0$, but I have a hard time to express that formally.
I'd be very grateful for a push in the right direction.
thanks for helping :)
real-analysis sequences-and-series convergence
asked Nov 29 '18 at 20:13
DDevelopsDDevelops
533
add a comment |
$(a_n)_{n in mathbb{N}},(b_n)_{n in mathbb{N}}$ are positive, real sequences!
Since $sum a_n$ converges, we know $limlimits_{nrightarrow infty}a_n =0$.
If $limlimits_{nrightarrow infty}b_n ne 0$ then $ liminflimits_{nrightarrow infty} frac{a_n}{b_n}=0$ is rather easy to show.
So let's assume $b_n$ converges to $0$. Since $sum a_n$ converges but $sum b_n$ does not, $a_n$ must somehow "converge faster" to $0$ than $b_n$ does, thus causing $ liminflimits_{nrightarrow infty} frac{a_n}{b_n}=0$, but I have a hard time to express that formally.
I'd be very grateful for a push in the right direction.
thanks for helping :)
real-analysis sequences-and-series convergence
asked Nov 29 '18 at 20:13
DDevelopsDDevelops
533
add a comment |
$(a_n)_{n in mathbb{N}},(b_n)_{n in mathbb{N}}$ are positive, real sequences!
Since $sum a_n$ converges, we know $limlimits_{nrightarrow infty}a_n =0$.
If $limlimits_{nrightarrow infty}b_n ne 0$ then $ liminflimits_{nrightarrow infty} frac{a_n}{b_n}=0$ is rather easy to show.
So let's assume $b_n$ converges to $0$. Since $sum a_n$ converges but $sum b_n$ does not, $a_n$ must somehow "converge faster" to $0$ than $b_n$ does, thus causing $ liminflimits_{nrightarrow infty} frac{a_n}{b_n}=0$, but I have a hard time to express that formally.
I'd be very grateful for a push in the right direction.
thanks for helping :)
real-analysis sequences-and-series convergence
asked Nov 29 '18 at 20:13
DDevelopsDDevelops
533
$(a_n)_{n in mathbb{N}},(b_n)_{n in mathbb{N}}$ are positive, real sequences!
Since $sum a_n$ converges, we know $limlimits_{nrightarrow infty}a_n =0$.
If $limlimits_{nrightarrow infty}b_n ne 0$ then $ liminflimits_{nrightarrow infty} frac{a_n}{b_n}=0$ is rather easy to show.
So let's assume $b_n$ converges to $0$. Since $sum a_n$ converges but $sum b_n$ does not, $a_n$ must somehow "converge faster" to $0$ than $b_n$ does, thus causing $ liminflimits_{nrightarrow infty} frac{a_n}{b_n}=0$, but I have a hard time to express that formally.
I'd be very grateful for a push in the right direction.
thanks for helping :)
real-analysis sequences-and-series convergence
real-analysis sequences-and-series convergence
asked Nov 29 '18 at 20:13
DDevelopsDDevelops
533
asked Nov 29 '18 at 20:13
DDevelopsDDevelops
533
asked Nov 29 '18 at 20:13
DDevelopsDDevelops
533
asked Nov 29 '18 at 20:13
DDevelopsDDevelops
533
asked Nov 29 '18 at 20:13
DDevelopsDDevelops
533
533
add a comment |
add a comment |
1 Answer
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Hint: If $ c = liminf_{nto infty} frac{a_n}{b_n} > 0$ then
$$
a_n ge frac c2 b_n ge 0
$$
for all sufficiently large $n$. What does that tell about the convergence
of the series $sum a_n$ and $sum b_n$?
answered Nov 29 '18 at 20:20
Martin RMartin R
27.3k33254
thanks, your approach led me to a solution :)
– DDevelops
Nov 30 '18 at 14:43
add a comment |
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1 Answer
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1 Answer
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Hint: If $ c = liminf_{nto infty} frac{a_n}{b_n} > 0$ then
$$
a_n ge frac c2 b_n ge 0
$$
for all sufficiently large $n$. What does that tell about the convergence
of the series $sum a_n$ and $sum b_n$?
answered Nov 29 '18 at 20:20
Martin RMartin R
27.3k33254
thanks, your approach led me to a solution :)
– DDevelops
Nov 30 '18 at 14:43
add a comment |
Hint: If $ c = liminf_{nto infty} frac{a_n}{b_n} > 0$ then
$$
a_n ge frac c2 b_n ge 0
$$
for all sufficiently large $n$. What does that tell about the convergence
of the series $sum a_n$ and $sum b_n$?
answered Nov 29 '18 at 20:20
Martin RMartin R
27.3k33254
thanks, your approach led me to a solution :)
– DDevelops
Nov 30 '18 at 14:43
add a comment |
Hint: If $ c = liminf_{nto infty} frac{a_n}{b_n} > 0$ then
$$
a_n ge frac c2 b_n ge 0
$$
for all sufficiently large $n$. What does that tell about the convergence
of the series $sum a_n$ and $sum b_n$?
answered Nov 29 '18 at 20:20
Martin RMartin R
27.3k33254
Hint: If $ c = liminf_{nto infty} frac{a_n}{b_n} > 0$ then
$$
a_n ge frac c2 b_n ge 0
$$
for all sufficiently large $n$. What does that tell about the convergence
of the series $sum a_n$ and $sum b_n$?
answered Nov 29 '18 at 20:20
Martin RMartin R
27.3k33254
answered Nov 29 '18 at 20:20
Martin RMartin R
27.3k33254
answered Nov 29 '18 at 20:20
Martin RMartin R
27.3k33254
answered Nov 29 '18 at 20:20
Martin RMartin R
27.3k33254
27.3k33254
thanks, your approach led me to a solution :)
– DDevelops
Nov 30 '18 at 14:43
add a comment |
thanks, your approach led me to a solution :)
– DDevelops
Nov 30 '18 at 14:43
thanks, your approach led me to a solution :)
– DDevelops
Nov 30 '18 at 14:43
thanks, your approach led me to a solution :)
– DDevelops
Nov 30 '18 at 14:43
add a comment |
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