The radius of convergence of a given power series

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I am working on some problems and getting in the following trouble.



As my calculating, I need to find $lambda$ such that the following series diverges.



$$sum_k=1^inftyfrac1lambdafraclambda(2k)^2(2k+1)^4(2k-1)^2$$



My attempt is as the following:



$sum_k=1^inftyfrac1lambdafrac^2left(2kright)^4left(2k+1right)^2right]left(2kright)^2left(2k+1right)^2left(2k-1right)^2\
= frac1sum_k=1^inftyfrac1frac^2left(2kright)^4left(2k+1right)^2right]left(2kright)^2left(2k+1right)^2left(2k-1right)^2\$



Setting $|z|=frac1$.



Consider $sum_k=1^inftyfrac1lambdafrac^2left(2kright)^4left(2k+1right)^2right]left(2kright)^2left(2k+1right)^2left(2k-1right)^2\$ as $a_n$.



The radius of convergence is $frac1R=lim_krightarrow inftyfraca_n+1a_n$



I am not sure my method. Is it true or not? If not, please show me a way to find it.



Thank you in advance




calculus sequences-and-series power-series




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  • I think it helps to separate the two terms. Sum of two positive series diverges iff one of them does.
    – Kavi Rama Murthy
    Jul 29 at 12:03














up vote
0
down vote

favorite
1












I am working on some problems and getting in the following trouble.



As my calculating, I need to find $lambda$ such that the following series diverges.



$$sum_k=1^inftyfrac1lambdafraclambda(2k)^2(2k+1)^4(2k-1)^2$$



My attempt is as the following:



$sum_k=1^inftyfrac1lambdafrac^2left(2kright)^4left(2k+1right)^2right]left(2kright)^2left(2k+1right)^2left(2k-1right)^2\
= frac1sum_k=1^inftyfrac1frac^2left(2kright)^4left(2k+1right)^2right]left(2kright)^2left(2k+1right)^2left(2k-1right)^2\$



Setting $|z|=frac1$.



Consider $sum_k=1^inftyfrac1lambdafrac^2left(2kright)^4left(2k+1right)^2right]left(2kright)^2left(2k+1right)^2left(2k-1right)^2\$ as $a_n$.



The radius of convergence is $frac1R=lim_krightarrow inftyfraca_n+1a_n$



I am not sure my method. Is it true or not? If not, please show me a way to find it.



Thank you in advance




calculus sequences-and-series power-series




share|cite|improve this question





















  • I think it helps to separate the two terms. Sum of two positive series diverges iff one of them does.
    – Kavi Rama Murthy
    Jul 29 at 12:03












up vote
0
down vote

favorite
1









up vote
0
down vote

favorite
1






1





I am working on some problems and getting in the following trouble.



As my calculating, I need to find $lambda$ such that the following series diverges.



$$sum_k=1^inftyfrac1lambdafraclambda(2k)^2(2k+1)^4(2k-1)^2$$



My attempt is as the following:



$sum_k=1^inftyfrac1lambdafrac^2left(2kright)^4left(2k+1right)^2right]left(2kright)^2left(2k+1right)^2left(2k-1right)^2\
= frac1sum_k=1^inftyfrac1frac^2left(2kright)^4left(2k+1right)^2right]left(2kright)^2left(2k+1right)^2left(2k-1right)^2\$



Setting $|z|=frac1$.



Consider $sum_k=1^inftyfrac1lambdafrac^2left(2kright)^4left(2k+1right)^2right]left(2kright)^2left(2k+1right)^2left(2k-1right)^2\$ as $a_n$.



The radius of convergence is $frac1R=lim_krightarrow inftyfraca_n+1a_n$



I am not sure my method. Is it true or not? If not, please show me a way to find it.



Thank you in advance




calculus sequences-and-series power-series




share|cite|improve this question













I am working on some problems and getting in the following trouble.



As my calculating, I need to find $lambda$ such that the following series diverges.



$$sum_k=1^inftyfrac1lambdafraclambda(2k)^2(2k+1)^4(2k-1)^2$$



My attempt is as the following:



$sum_k=1^inftyfrac1lambdafrac^2left(2kright)^4left(2k+1right)^2right]left(2kright)^2left(2k+1right)^2left(2k-1right)^2\
= frac1sum_k=1^inftyfrac1frac^2left(2kright)^4left(2k+1right)^2right]left(2kright)^2left(2k+1right)^2left(2k-1right)^2\$



Setting $|z|=frac1$.



Consider $sum_k=1^inftyfrac1lambdafrac^2left(2kright)^4left(2k+1right)^2right]left(2kright)^2left(2k+1right)^2left(2k-1right)^2\$ as $a_n$.



The radius of convergence is $frac1R=lim_krightarrow inftyfraca_n+1a_n$



I am not sure my method. Is it true or not? If not, please show me a way to find it.



Thank you in advance





calculus sequences-and-series power-series





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edited Aug 2 at 1:47









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  • I think it helps to separate the two terms. Sum of two positive series diverges iff one of them does.
    – Kavi Rama Murthy
    Jul 29 at 12:03
















  • I think it helps to separate the two terms. Sum of two positive series diverges iff one of them does.
    – Kavi Rama Murthy
    Jul 29 at 12:03















I think it helps to separate the two terms. Sum of two positive series diverges iff one of them does.
– Kavi Rama Murthy
Jul 29 at 12:03




I think it helps to separate the two terms. Sum of two positive series diverges iff one of them does.
– Kavi Rama Murthy
Jul 29 at 12:03















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