Dgdrxrt

Multi tool use

A newer friend of my brother's gave him a load of baseball cards that are supposedly extremely valuable. Is this a scam?

Languages that we cannot (dis)prove to be Context-Free

What are the differences between the usage of 'it' and 'they'?

How much RAM could one put in a typical 80386 setup?

Did Shadowfax go to Valinor?

can i play a electric guitar through a bass amp?

Is it important to consider tone, melody, and musical form while writing a song?

What is the offset in a seaplane's hull?

Today is the Center

Approximately how much travel time was saved by the opening of the Suez Canal in 1869?

Modeling an IPv4 Address

Can I make popcorn with any corn?

Why do falling prices hurt debtors?

How do I create uniquely male characters?

Why are 150k or 200k jobs considered good when there are 300k+ births a month?

What do the dots in this tr command do: tr .............A-Z A-ZA-Z <<< "JVPQBOV" (with 13 dots)

Is it unprofessional to ask if a job posting on GlassDoor is real?

Can a Warlock become Neutral Good?

Why can't I see bouncing of a switch on an oscilloscope?

In Japanese, what’s the difference between “Tonari ni” (となりに) and “Tsugi” (つぎ)? When would you use one over the other?

Why Is Death Allowed In the Matrix?

How to write a macro that is braces sensitive?

How does one intimidate enemies without having the capacity for violence?

Do VLANs within a subnet need to have their own subnet for router on a stick?

Showing that $sum_n=1^inftyfraca_na_n+b_n$ converges. [duplicate]

how prove $sum_n=1^inftyfraca_nb_n+a_n $is convergent?prove series converges$frac a_n+1a_n le frac b_n+1b_n$ If $sum_n=1^infty b_n$ converges then $sum_n=1^infty a_n$ converges as wellIf $sum a_n$ converges and $b_n=sumlimits_k=n^inftya_n $, prove that $sum fraca_nb_n$ divergesIf $sum a_n b_n$ converges for all $(b_n)$ such that $b_n to 0$, then $sum |a_n|$ converges.$sumlimits_n=1^infty a_n^2$ and $sumlimits_n=1^infty b_n^2$ converge show $sumlimits_n=1^infty a_n b_n$ converges absolutelyProve if $sumlimits_n=1^ infty a_n$ converges, $b_n$ is bounded & monotone, then $sumlimits_n=1^ infty a_nb_n$ converges.If $sum_n=0^infty|a_n|^p,sum_n=0^infty|b_n|^p $ converge then $sum_n=0^infty|a_n+b_n|^p$ convergesA question about real series $sum_n=1^infty a_n$ and $sum_n=1^infty b_n$Show that $sum_n=0^infty(sum_j=0^n a_jb_n-j)$ converges to $(sum_n=0^inftyb_n)(sum_n=0^inftya_n)$.$sum_n=1^infty a_n^b_n$ convergesProve $(a_n,b_n >0) land sum a_n $ converges $ land sum b_n $ diverges$implies liminflimits_nrightarrow infty fraca_nb_n=0$

6

1

$begingroup$

This question already has an answer here:

• 
  how prove $sum_n=1^inftyfraca_nb_n+a_n $is convergent?
  
  4 answers
  
  

Show that if $a_n,b_ninmathbbR$, $(a_n+b_n)b_nneq0$ and both $displaystylesum_n=1^inftyfraca_nb_n$ and $displaystylesum_n=1^inftyleft(fraca_nb_nright)^2$ converge, then $displaystylesum_n=1^inftyfraca_na_n+b_n$ converges.

If $a_n$ is positive, I have been able to solve. How we can solve in general?

real-analysis sequences-and-series

share|cite|improve this question

edited Mar 29 at 20:23

TheSimpliFire

13k62464

asked Mar 29 at 17:07

J.DoeJ.Doe

943

$endgroup$

marked as duplicate by Martin R, Lord Shark the Unknown, FredH, Jyrki Lahtonen, Leucippus Mar 30 at 6:33

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Also: math.stackexchange.com/q/2154959/42969.
  $endgroup$
  – Martin R
  Mar 30 at 2:31
  

  
  
  
  

add a comment | 

6

1

$begingroup$

This question already has an answer here:

• 
  how prove $sum_n=1^inftyfraca_nb_n+a_n $is convergent?
  
  4 answers
  
  

Show that if $a_n,b_ninmathbbR$, $(a_n+b_n)b_nneq0$ and both $displaystylesum_n=1^inftyfraca_nb_n$ and $displaystylesum_n=1^inftyleft(fraca_nb_nright)^2$ converge, then $displaystylesum_n=1^inftyfraca_na_n+b_n$ converges.

If $a_n$ is positive, I have been able to solve. How we can solve in general?

real-analysis sequences-and-series

share|cite|improve this question

edited Mar 29 at 20:23

TheSimpliFire

13k62464

asked Mar 29 at 17:07

J.DoeJ.Doe

943

$endgroup$

marked as duplicate by Martin R, Lord Shark the Unknown, FredH, Jyrki Lahtonen, Leucippus Mar 30 at 6:33

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Also: math.stackexchange.com/q/2154959/42969.
  $endgroup$
  – Martin R
  Mar 30 at 2:31
  

  
  
  
  

add a comment | 

$begingroup$

This question already has an answer here:

• 
  how prove $sum_n=1^inftyfraca_nb_n+a_n $is convergent?
  
  4 answers
  
  

Show that if $a_n,b_ninmathbbR$, $(a_n+b_n)b_nneq0$ and both $displaystylesum_n=1^inftyfraca_nb_n$ and $displaystylesum_n=1^inftyleft(fraca_nb_nright)^2$ converge, then $displaystylesum_n=1^inftyfraca_na_n+b_n$ converges.

If $a_n$ is positive, I have been able to solve. How we can solve in general?

real-analysis sequences-and-series

share|cite|improve this question

edited Mar 29 at 20:23

TheSimpliFire

13k62464

asked Mar 29 at 17:07

J.DoeJ.Doe

943

$endgroup$

share|cite|improve this question

edited Mar 29 at 20:23

TheSimpliFire

13k62464

asked Mar 29 at 17:07

J.DoeJ.Doe

943

share|cite|improve this question

edited Mar 29 at 20:23

TheSimpliFire

13k62464

asked Mar 29 at 17:07

J.DoeJ.Doe

943

edited Mar 29 at 20:23

TheSimpliFire

13k62464

edited Mar 29 at 20:23

TheSimpliFire

13k62464

asked Mar 29 at 17:07

J.DoeJ.Doe

943

asked Mar 29 at 17:07

J.DoeJ.Doe

943

1 Answer
1

active

oldest

votes

9

$begingroup$

Write $c_n=fraca_nb_n$. Then we have $c_nne -1$, and also $sum c_n$, $sum c_n^2$ converge. We need to show $sum fracc_n1+c_n$ converges.

It suffices to show that the sum of
$$c_n-fracc_n1+c_n=fracc_n^21+c_n.$$
converges, since $sum c_n$ converges.

But $1+c_nto 1$. Then $sumfracc_n^21+c_n$ converges by comparison to $sum c_n^2 $.

share|cite|improve this answer

answered Mar 29 at 17:19

Eclipse SunEclipse Sun

8,0151438

$endgroup$

add a comment | 

9

$begingroup$

Write $c_n=fraca_nb_n$. Then we have $c_nne -1$, and also $sum c_n$, $sum c_n^2$ converge. We need to show $sum fracc_n1+c_n$ converges.

It suffices to show that the sum of
$$c_n-fracc_n1+c_n=fracc_n^21+c_n.$$
converges, since $sum c_n$ converges.

But $1+c_nto 1$. Then $sumfracc_n^21+c_n$ converges by comparison to $sum c_n^2 $.

share|cite|improve this answer

answered Mar 29 at 17:19

Eclipse SunEclipse Sun

8,0151438

$endgroup$

add a comment | 

9

$begingroup$

Write $c_n=fraca_nb_n$. Then we have $c_nne -1$, and also $sum c_n$, $sum c_n^2$ converge. We need to show $sum fracc_n1+c_n$ converges.

It suffices to show that the sum of
$$c_n-fracc_n1+c_n=fracc_n^21+c_n.$$
converges, since $sum c_n$ converges.

But $1+c_nto 1$. Then $sumfracc_n^21+c_n$ converges by comparison to $sum c_n^2 $.

share|cite|improve this answer

answered Mar 29 at 17:19

Eclipse SunEclipse Sun

8,0151438

$endgroup$

add a comment | 

$begingroup$

Write $c_n=fraca_nb_n$. Then we have $c_nne -1$, and also $sum c_n$, $sum c_n^2$ converge. We need to show $sum fracc_n1+c_n$ converges.

It suffices to show that the sum of
$$c_n-fracc_n1+c_n=fracc_n^21+c_n.$$
converges, since $sum c_n$ converges.

But $1+c_nto 1$. Then $sumfracc_n^21+c_n$ converges by comparison to $sum c_n^2 $.

share|cite|improve this answer

answered Mar 29 at 17:19

Eclipse SunEclipse Sun

8,0151438

$endgroup$

share|cite|improve this answer

answered Mar 29 at 17:19

Eclipse SunEclipse Sun

8,0151438

answered Mar 29 at 17:19

Eclipse SunEclipse Sun

8,0151438

answered Mar 29 at 17:19

Eclipse SunEclipse Sun

8,0151438