Show that $sum^infty_n=1(-1)^n+1dfrac1n+x^4$ is uniformly convergent on $BbbR$
Clash Royale CLAN TAG#URR8PPP
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Show that the following series is uniformly convergent on $BbbR$
beginalignsum^infty_n=1(-1)^n+1dfrac1n+x^4endalign
MY TRIAL
I tried using the alternating series test before the $beta_n$ approach.
Let $f_n(x)=dfrac1n+x^4,;forall;xinBbbR,;ninBbbN,$
then
- $f_n(x)=dfrac1n+x^4geq 0$
- $f_n+1(x)leq f_n(x)$
- $f_n(x)=dfrac1n+x^4to 0$
Then,
beginalignbeta_n &=suplimits_xinBbbRleft|s_n(x)-sum^infty_i=1f_i(x)right|\&=suplimits_xinBbbRleft|sum^n_i=1(-1)^i+1f_i(x)-sum^infty_i=1(-1)^i+1f_i(x)right|\&=suplimits_xinBbbRleft|sum^n_i=1(-1)^i+1dfrac1i+x^4-sum^infty_i=1(-1)^i+1dfrac1i+x^4right|\&=suplimits_xinBbbRleft|(-1)^n+2dfrac1(n+1)+x^4+(-1)^n+3dfrac1(n+2)+x^4+(-1)^n+4dfrac1(n+3)+x^4cdotsright|\&=suplimits_xinBbbRleft|dfrac1(n+1)+x^4-dfrac1(n+2)+x^4+dfrac1(n+3)+x^4-dfrac1(n+4)+x^4cdotsright|\&=suplimits_xinBbbRleft|dfrac1(n+1)+x^4-left(dfrac1(n+2)+x^4-dfrac1(n+3)+x^4right)-left(dfrac1(n+4)+x^4-dfrac1(n+5)+x^4right)cdotsright|\&leq suplimits_xinBbbRleft|dfrac1(n+1)+x^4right|to 0,;;textas;ntoinftyendalign
and we are done!
Kindly help me check if I'm correct! Constructive criticisms will be highly welcome! I'll also love to see other approaches to this problem. Thanks!
real-analysis sequences-and-series analysis convergence uniform-convergence
asked Sep 5 at 10:29
Micheal
25510
add a comment |Â
up vote
0
down vote
favorite
Show that the following series is uniformly convergent on $BbbR$
beginalignsum^infty_n=1(-1)^n+1dfrac1n+x^4endalign
MY TRIAL
I tried using the alternating series test before the $beta_n$ approach.
Let $f_n(x)=dfrac1n+x^4,;forall;xinBbbR,;ninBbbN,$
then
- $f_n(x)=dfrac1n+x^4geq 0$
- $f_n+1(x)leq f_n(x)$
- $f_n(x)=dfrac1n+x^4to 0$
Then,
beginalignbeta_n &=suplimits_xinBbbRleft|s_n(x)-sum^infty_i=1f_i(x)right|\&=suplimits_xinBbbRleft|sum^n_i=1(-1)^i+1f_i(x)-sum^infty_i=1(-1)^i+1f_i(x)right|\&=suplimits_xinBbbRleft|sum^n_i=1(-1)^i+1dfrac1i+x^4-sum^infty_i=1(-1)^i+1dfrac1i+x^4right|\&=suplimits_xinBbbRleft|(-1)^n+2dfrac1(n+1)+x^4+(-1)^n+3dfrac1(n+2)+x^4+(-1)^n+4dfrac1(n+3)+x^4cdotsright|\&=suplimits_xinBbbRleft|dfrac1(n+1)+x^4-dfrac1(n+2)+x^4+dfrac1(n+3)+x^4-dfrac1(n+4)+x^4cdotsright|\&=suplimits_xinBbbRleft|dfrac1(n+1)+x^4-left(dfrac1(n+2)+x^4-dfrac1(n+3)+x^4right)-left(dfrac1(n+4)+x^4-dfrac1(n+5)+x^4right)cdotsright|\&leq suplimits_xinBbbRleft|dfrac1(n+1)+x^4right|to 0,;;textas;ntoinftyendalign
and we are done!
Kindly help me check if I'm correct! Constructive criticisms will be highly welcome! I'll also love to see other approaches to this problem. Thanks!
real-analysis sequences-and-series analysis convergence uniform-convergence
asked Sep 5 at 10:29
Micheal
25510
Hint: Abel-Dirichlet tests.
– xbh
Sep 5 at 10:34
Maybe you could use the same technique to estimate the $sup$.
– xbh
Sep 5 at 10:36
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Show that the following series is uniformly convergent on $BbbR$
beginalignsum^infty_n=1(-1)^n+1dfrac1n+x^4endalign
MY TRIAL
I tried using the alternating series test before the $beta_n$ approach.
Let $f_n(x)=dfrac1n+x^4,;forall;xinBbbR,;ninBbbN,$
then
- $f_n(x)=dfrac1n+x^4geq 0$
- $f_n+1(x)leq f_n(x)$
- $f_n(x)=dfrac1n+x^4to 0$
Then,
beginalignbeta_n &=suplimits_xinBbbRleft|s_n(x)-sum^infty_i=1f_i(x)right|\&=suplimits_xinBbbRleft|sum^n_i=1(-1)^i+1f_i(x)-sum^infty_i=1(-1)^i+1f_i(x)right|\&=suplimits_xinBbbRleft|sum^n_i=1(-1)^i+1dfrac1i+x^4-sum^infty_i=1(-1)^i+1dfrac1i+x^4right|\&=suplimits_xinBbbRleft|(-1)^n+2dfrac1(n+1)+x^4+(-1)^n+3dfrac1(n+2)+x^4+(-1)^n+4dfrac1(n+3)+x^4cdotsright|\&=suplimits_xinBbbRleft|dfrac1(n+1)+x^4-dfrac1(n+2)+x^4+dfrac1(n+3)+x^4-dfrac1(n+4)+x^4cdotsright|\&=suplimits_xinBbbRleft|dfrac1(n+1)+x^4-left(dfrac1(n+2)+x^4-dfrac1(n+3)+x^4right)-left(dfrac1(n+4)+x^4-dfrac1(n+5)+x^4right)cdotsright|\&leq suplimits_xinBbbRleft|dfrac1(n+1)+x^4right|to 0,;;textas;ntoinftyendalign
and we are done!
Kindly help me check if I'm correct! Constructive criticisms will be highly welcome! I'll also love to see other approaches to this problem. Thanks!
real-analysis sequences-and-series analysis convergence uniform-convergence
asked Sep 5 at 10:29
Micheal
25510
Show that the following series is uniformly convergent on $BbbR$
beginalignsum^infty_n=1(-1)^n+1dfrac1n+x^4endalign
MY TRIAL
I tried using the alternating series test before the $beta_n$ approach.
Let $f_n(x)=dfrac1n+x^4,;forall;xinBbbR,;ninBbbN,$
then
- $f_n(x)=dfrac1n+x^4geq 0$
- $f_n+1(x)leq f_n(x)$
- $f_n(x)=dfrac1n+x^4to 0$
Then,
beginalignbeta_n &=suplimits_xinBbbRleft|s_n(x)-sum^infty_i=1f_i(x)right|\&=suplimits_xinBbbRleft|sum^n_i=1(-1)^i+1f_i(x)-sum^infty_i=1(-1)^i+1f_i(x)right|\&=suplimits_xinBbbRleft|sum^n_i=1(-1)^i+1dfrac1i+x^4-sum^infty_i=1(-1)^i+1dfrac1i+x^4right|\&=suplimits_xinBbbRleft|(-1)^n+2dfrac1(n+1)+x^4+(-1)^n+3dfrac1(n+2)+x^4+(-1)^n+4dfrac1(n+3)+x^4cdotsright|\&=suplimits_xinBbbRleft|dfrac1(n+1)+x^4-dfrac1(n+2)+x^4+dfrac1(n+3)+x^4-dfrac1(n+4)+x^4cdotsright|\&=suplimits_xinBbbRleft|dfrac1(n+1)+x^4-left(dfrac1(n+2)+x^4-dfrac1(n+3)+x^4right)-left(dfrac1(n+4)+x^4-dfrac1(n+5)+x^4right)cdotsright|\&leq suplimits_xinBbbRleft|dfrac1(n+1)+x^4right|to 0,;;textas;ntoinftyendalign
and we are done!
Kindly help me check if I'm correct! Constructive criticisms will be highly welcome! I'll also love to see other approaches to this problem. Thanks!
real-analysis sequences-and-series analysis convergence uniform-convergence
real-analysis sequences-and-series analysis convergence uniform-convergence
asked Sep 5 at 10:29
Micheal
25510
asked Sep 5 at 10:29
Micheal
25510
edited Sep 5 at 11:08
asked Sep 5 at 10:29
Micheal
25510
asked Sep 5 at 10:29
Micheal
25510
asked Sep 5 at 10:29
Micheal
25510
25510
Hint: Abel-Dirichlet tests.
– xbh
Sep 5 at 10:34
Maybe you could use the same technique to estimate the $sup$.
– xbh
Sep 5 at 10:36
add a comment |Â
Hint: Abel-Dirichlet tests.
– xbh
Sep 5 at 10:34
Maybe you could use the same technique to estimate the $sup$.
– xbh
Sep 5 at 10:36
Hint: Abel-Dirichlet tests.
– xbh
Sep 5 at 10:34
Hint: Abel-Dirichlet tests.
– xbh
Sep 5 at 10:34
Maybe you could use the same technique to estimate the $sup$.
– xbh
Sep 5 at 10:36
Maybe you could use the same technique to estimate the $sup$.
– xbh
Sep 5 at 10:36
add a comment |Â
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Hint: Abel-Dirichlet tests.
– xbh
Sep 5 at 10:34
Maybe you could use the same technique to estimate the $sup$.
– xbh
Sep 5 at 10:36