Alternating series, $sum_n=1^infty n^2 a_n^2$ conv. $implies$ $sum_n=1^infty (-1)^n a_n$ conv.Leibniz's alternating series testConvergence of an alternating series (from exam Q)If $sum|a_n|<infty$ and $sum |b_n|<infty$, is the sum $sum_k=0^infty sum_n=0^infty |a_n||b_n+k|<infty$?If $sum_n_0^infty a_n$ diverges prove that $sum_n_0^infty fraca_na_1+a_2+…+a_n = +infty $Do the following series converge if $a_n>0$ and $ sum_n=1^inftya_n$ diverges?$sum_n= 0^inftya_n$ converges, what other series must then also converge?Contradictions between the Alternating Series Test & Divergence Test?If $sum_n=0^infty a_n$ converges absolutely, so does $sum_n=0^infty fraca_n^22-a_n^2$Counterexample to Leibniz criterion for alternating seriesExample of series such that every $sum_i=1^infty b_n_i$ converges but $sum_n=1^infty |b_n|$ diverges
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Alternating series, $sum_n=1^infty n^2 a_n^2$ conv. $implies$ $sum_n=1^infty (-1)^n a_n$ conv.
Leibniz's alternating series testConvergence of an alternating series (from exam Q)If $sum|a_n|<infty$ and $sum |b_n|<infty$, is the sum $sum_k=0^infty sum_n=0^infty |a_n||b_n+k|<infty$?If $sum_n_0^infty a_n$ diverges prove that $sum_n_0^infty fraca_na_1+a_2+…+a_n = +infty $Do the following series converge if $a_n>0$ and $ sum_n=1^inftya_n$ diverges?$sum_n= 0^inftya_n$ converges, what other series must then also converge?Contradictions between the Alternating Series Test & Divergence Test?If $sum_n=0^infty a_n$ converges absolutely, so does $sum_n=0^infty fraca_n^22-a_n^2$Counterexample to Leibniz criterion for alternating seriesExample of series such that every $sum_i=1^infty b_n_i$ converges but $sum_n=1^infty |b_n|$ diverges
$begingroup$
Suppose that $ sum_n=1^infty n^2 a_n^2 $ converges. Does this imply $ sum_n=1^infty (-1)^n a_n $ converges?
I guess it converges, but I couldn't prove it.
I have tried writing $b_n = na_n$ but it seems that the alternating series test did not work.
Any help would be greatly appreciated.
real-analysis sequences-and-series
asked Mar 15 at 2:59
FuroCharuFuroCharu
2715
$endgroup$
add a comment |
$begingroup$
Suppose that $ sum_n=1^infty n^2 a_n^2 $ converges. Does this imply $ sum_n=1^infty (-1)^n a_n $ converges?
I guess it converges, but I couldn't prove it.
I have tried writing $b_n = na_n$ but it seems that the alternating series test did not work.
Any help would be greatly appreciated.
real-analysis sequences-and-series
asked Mar 15 at 2:59
FuroCharuFuroCharu
2715
$endgroup$
1
$begingroup$
Cauchy Schwarz$ $
$endgroup$
– reuns
Mar 15 at 3:06
add a comment |
$begingroup$
Suppose that $ sum_n=1^infty n^2 a_n^2 $ converges. Does this imply $ sum_n=1^infty (-1)^n a_n $ converges?
I guess it converges, but I couldn't prove it.
I have tried writing $b_n = na_n$ but it seems that the alternating series test did not work.
Any help would be greatly appreciated.
real-analysis sequences-and-series
asked Mar 15 at 2:59
FuroCharuFuroCharu
2715
$endgroup$
Suppose that $ sum_n=1^infty n^2 a_n^2 $ converges. Does this imply $ sum_n=1^infty (-1)^n a_n $ converges?
I guess it converges, but I couldn't prove it.
I have tried writing $b_n = na_n$ but it seems that the alternating series test did not work.
Any help would be greatly appreciated.
real-analysis sequences-and-series
real-analysis sequences-and-series
asked Mar 15 at 2:59
FuroCharuFuroCharu
2715
asked Mar 15 at 2:59
FuroCharuFuroCharu
2715
asked Mar 15 at 2:59
FuroCharuFuroCharu
2715
asked Mar 15 at 2:59
FuroCharuFuroCharu
2715
asked Mar 15 at 2:59
FuroCharuFuroCharu
2715
2715
1
$begingroup$
Cauchy Schwarz$ $
$endgroup$
– reuns
Mar 15 at 3:06
add a comment |
1
$begingroup$
Cauchy Schwarz$ $
$endgroup$
– reuns
Mar 15 at 3:06
1
1
$begingroup$
Cauchy Schwarz$ $
$endgroup$
– reuns
Mar 15 at 3:06
$begingroup$
Cauchy Schwarz$ $
$endgroup$
– reuns
Mar 15 at 3:06
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Hint:
beginalign
left|sum_n=K^N (-1)^na_nright| leq left(sum^N_n=K frac1n^2right)^1/2 left(sum^N_n=K n^2a_n^2 right)^1/2
endalign
answered Mar 15 at 3:07
Jacky ChongJacky Chong
19.4k21129
$endgroup$
$begingroup$
Thanks, I feel so dumb now I forgot to try using cauchy schwarz.
$endgroup$
– FuroCharu
Mar 15 at 3:09
add a comment |
Your Answer
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1 Answer
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oldest
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1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Hint:
beginalign
left|sum_n=K^N (-1)^na_nright| leq left(sum^N_n=K frac1n^2right)^1/2 left(sum^N_n=K n^2a_n^2 right)^1/2
endalign
answered Mar 15 at 3:07
Jacky ChongJacky Chong
19.4k21129
$endgroup$
$begingroup$
Thanks, I feel so dumb now I forgot to try using cauchy schwarz.
$endgroup$
– FuroCharu
Mar 15 at 3:09
add a comment |
$begingroup$
Hint:
beginalign
left|sum_n=K^N (-1)^na_nright| leq left(sum^N_n=K frac1n^2right)^1/2 left(sum^N_n=K n^2a_n^2 right)^1/2
endalign
answered Mar 15 at 3:07
Jacky ChongJacky Chong
19.4k21129
$endgroup$
$begingroup$
Thanks, I feel so dumb now I forgot to try using cauchy schwarz.
$endgroup$
– FuroCharu
Mar 15 at 3:09
add a comment |
$begingroup$
Hint:
beginalign
left|sum_n=K^N (-1)^na_nright| leq left(sum^N_n=K frac1n^2right)^1/2 left(sum^N_n=K n^2a_n^2 right)^1/2
endalign
answered Mar 15 at 3:07
Jacky ChongJacky Chong
19.4k21129
$endgroup$
Hint:
beginalign
left|sum_n=K^N (-1)^na_nright| leq left(sum^N_n=K frac1n^2right)^1/2 left(sum^N_n=K n^2a_n^2 right)^1/2
endalign
answered Mar 15 at 3:07
Jacky ChongJacky Chong
19.4k21129
answered Mar 15 at 3:07
Jacky ChongJacky Chong
19.4k21129
answered Mar 15 at 3:07
Jacky ChongJacky Chong
19.4k21129
answered Mar 15 at 3:07
Jacky ChongJacky Chong
19.4k21129
19.4k21129
$begingroup$
Thanks, I feel so dumb now I forgot to try using cauchy schwarz.
$endgroup$
– FuroCharu
Mar 15 at 3:09
add a comment |
$begingroup$
Thanks, I feel so dumb now I forgot to try using cauchy schwarz.
$endgroup$
– FuroCharu
Mar 15 at 3:09
$begingroup$
Thanks, I feel so dumb now I forgot to try using cauchy schwarz.
$endgroup$
– FuroCharu
Mar 15 at 3:09
$begingroup$
Thanks, I feel so dumb now I forgot to try using cauchy schwarz.
$endgroup$
– FuroCharu
Mar 15 at 3:09
add a comment |
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$begingroup$
Cauchy Schwarz$ $
$endgroup$
– reuns
Mar 15 at 3:06