Evaluating the series $sum_k=1^infty frac2times 3^k4^2k+1$

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$$sum_k=1^infty frac2times 3^k4^2k+1$$



Hi all, I finally am getting the hang of MathJax (sort of) thank goodness! I was hoping for some help on a problem involving series. I am stuck trying to re-write this problem to make it easier to solve. For example, I was hoping to use the fact that a Sum = $$Sinfty= fraca_11-r$$

to solve but unlike a problem with simply k+1 in the denominator, this has a constant infront of it which is throwing me off.. any tips on how approach solving or a trick that I am unaware of?



Thank you!




calculus sequences-and-series geometric-progressions




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    Next stop: getting the hang of titles! :) The key directive is that people should be able to know something about your question without having to open it.
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  • Thank you Asaf, had no idea we could directly integrate problems in the title and have mathjax pick it up!
    – jackbenimbo
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up vote
2
down vote

favorite












$$sum_k=1^infty frac2times 3^k4^2k+1$$



Hi all, I finally am getting the hang of MathJax (sort of) thank goodness! I was hoping for some help on a problem involving series. I am stuck trying to re-write this problem to make it easier to solve. For example, I was hoping to use the fact that a Sum = $$Sinfty= fraca_11-r$$

to solve but unlike a problem with simply k+1 in the denominator, this has a constant infront of it which is throwing me off.. any tips on how approach solving or a trick that I am unaware of?



Thank you!




calculus sequences-and-series geometric-progressions




share|cite|improve this question

















  • 1




    Next stop: getting the hang of titles! :) The key directive is that people should be able to know something about your question without having to open it.
    – Asaf Karagila
    yesterday










  • Thank you Asaf, had no idea we could directly integrate problems in the title and have mathjax pick it up!
    – jackbenimbo
    14 hours ago












up vote
2
down vote

favorite









up vote
2
down vote

favorite











$$sum_k=1^infty frac2times 3^k4^2k+1$$



Hi all, I finally am getting the hang of MathJax (sort of) thank goodness! I was hoping for some help on a problem involving series. I am stuck trying to re-write this problem to make it easier to solve. For example, I was hoping to use the fact that a Sum = $$Sinfty= fraca_11-r$$

to solve but unlike a problem with simply k+1 in the denominator, this has a constant infront of it which is throwing me off.. any tips on how approach solving or a trick that I am unaware of?



Thank you!




calculus sequences-and-series geometric-progressions




share|cite|improve this question













$$sum_k=1^infty frac2times 3^k4^2k+1$$



Hi all, I finally am getting the hang of MathJax (sort of) thank goodness! I was hoping for some help on a problem involving series. I am stuck trying to re-write this problem to make it easier to solve. For example, I was hoping to use the fact that a Sum = $$Sinfty= fraca_11-r$$

to solve but unlike a problem with simply k+1 in the denominator, this has a constant infront of it which is throwing me off.. any tips on how approach solving or a trick that I am unaware of?



Thank you!





calculus sequences-and-series geometric-progressions





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited yesterday









Martin Sleziak

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43.4k6111259









asked yesterday









jackbenimbo

446




446







  • 1




    Next stop: getting the hang of titles! :) The key directive is that people should be able to know something about your question without having to open it.
    – Asaf Karagila
    yesterday










  • Thank you Asaf, had no idea we could directly integrate problems in the title and have mathjax pick it up!
    – jackbenimbo
    14 hours ago












  • 1




    Next stop: getting the hang of titles! :) The key directive is that people should be able to know something about your question without having to open it.
    – Asaf Karagila
    yesterday










  • Thank you Asaf, had no idea we could directly integrate problems in the title and have mathjax pick it up!
    – jackbenimbo
    14 hours ago







1




1




Next stop: getting the hang of titles! :) The key directive is that people should be able to know something about your question without having to open it.
– Asaf Karagila
yesterday




Next stop: getting the hang of titles! :) The key directive is that people should be able to know something about your question without having to open it.
– Asaf Karagila
yesterday












Thank you Asaf, had no idea we could directly integrate problems in the title and have mathjax pick it up!
– jackbenimbo
14 hours ago




Thank you Asaf, had no idea we could directly integrate problems in the title and have mathjax pick it up!
– jackbenimbo
14 hours ago










1 Answer
1






active

oldest

votes

















up vote
7
down vote



accepted










This is a geometric series
$$sum_k=1^infty frac2*3^k4^2k+1=sum_k=1^infty dfrac12left(frac316right)^k$$
with $a_1=dfrac12dfrac316$ and $q=dfrac316$, then
$$S_infty=dfracdfrac12dfrac3161-dfrac316=dfrac326$$






share|cite|improve this answer





















  • so you reduced 2/4, how are you getting 3/16 though? (Thanks in advance)
    – jackbenimbo
    yesterday










  • the power $4$ is $2k+1$, $(4^2)^k*4$.
    – user 108128
    yesterday










  • I see it clearly now thank you so much for your time on a Saturday evening! Cheers
    – jackbenimbo
    yesterday










  • you are welcome.
    – user 108128
    yesterday










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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
7
down vote



accepted










This is a geometric series
$$sum_k=1^infty frac2*3^k4^2k+1=sum_k=1^infty dfrac12left(frac316right)^k$$
with $a_1=dfrac12dfrac316$ and $q=dfrac316$, then
$$S_infty=dfracdfrac12dfrac3161-dfrac316=dfrac326$$






share|cite|improve this answer





















  • so you reduced 2/4, how are you getting 3/16 though? (Thanks in advance)
    – jackbenimbo
    yesterday










  • the power $4$ is $2k+1$, $(4^2)^k*4$.
    – user 108128
    yesterday










  • I see it clearly now thank you so much for your time on a Saturday evening! Cheers
    – jackbenimbo
    yesterday










  • you are welcome.
    – user 108128
    yesterday














up vote
7
down vote



accepted










This is a geometric series
$$sum_k=1^infty frac2*3^k4^2k+1=sum_k=1^infty dfrac12left(frac316right)^k$$
with $a_1=dfrac12dfrac316$ and $q=dfrac316$, then
$$S_infty=dfracdfrac12dfrac3161-dfrac316=dfrac326$$






share|cite|improve this answer





















  • so you reduced 2/4, how are you getting 3/16 though? (Thanks in advance)
    – jackbenimbo
    yesterday










  • the power $4$ is $2k+1$, $(4^2)^k*4$.
    – user 108128
    yesterday










  • I see it clearly now thank you so much for your time on a Saturday evening! Cheers
    – jackbenimbo
    yesterday










  • you are welcome.
    – user 108128
    yesterday












up vote
7
down vote



accepted







up vote
7
down vote



accepted






This is a geometric series
$$sum_k=1^infty frac2*3^k4^2k+1=sum_k=1^infty dfrac12left(frac316right)^k$$
with $a_1=dfrac12dfrac316$ and $q=dfrac316$, then
$$S_infty=dfracdfrac12dfrac3161-dfrac316=dfrac326$$






share|cite|improve this answer













This is a geometric series
$$sum_k=1^infty frac2*3^k4^2k+1=sum_k=1^infty dfrac12left(frac316right)^k$$
with $a_1=dfrac12dfrac316$ and $q=dfrac316$, then
$$S_infty=dfracdfrac12dfrac3161-dfrac316=dfrac326$$







share|cite|improve this answer













share|cite|improve this answer



share|cite|improve this answer











answered yesterday









user 108128

18.6k41544




18.6k41544











  • so you reduced 2/4, how are you getting 3/16 though? (Thanks in advance)
    – jackbenimbo
    yesterday










  • the power $4$ is $2k+1$, $(4^2)^k*4$.
    – user 108128
    yesterday










  • I see it clearly now thank you so much for your time on a Saturday evening! Cheers
    – jackbenimbo
    yesterday










  • you are welcome.
    – user 108128
    yesterday
















  • so you reduced 2/4, how are you getting 3/16 though? (Thanks in advance)
    – jackbenimbo
    yesterday










  • the power $4$ is $2k+1$, $(4^2)^k*4$.
    – user 108128
    yesterday










  • I see it clearly now thank you so much for your time on a Saturday evening! Cheers
    – jackbenimbo
    yesterday










  • you are welcome.
    – user 108128
    yesterday















so you reduced 2/4, how are you getting 3/16 though? (Thanks in advance)
– jackbenimbo
yesterday




so you reduced 2/4, how are you getting 3/16 though? (Thanks in advance)
– jackbenimbo
yesterday












the power $4$ is $2k+1$, $(4^2)^k*4$.
– user 108128
yesterday




the power $4$ is $2k+1$, $(4^2)^k*4$.
– user 108128
yesterday












I see it clearly now thank you so much for your time on a Saturday evening! Cheers
– jackbenimbo
yesterday




I see it clearly now thank you so much for your time on a Saturday evening! Cheers
– jackbenimbo
yesterday












you are welcome.
– user 108128
yesterday




you are welcome.
– user 108128
yesterday












 

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