Consider the series $sum_{n=1}^infty frac{(-1)^n}{n+x}$
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Consider the series
$$sum_{n=1}^infty frac{(-1)^n}{n+x}$$
Find all $x in mathbb{R}$ at which the series converges. Converges absolutely. Find all intervals of $mathbb{R}$ where the series defining $f$ converges uniformly, and all intervals of $mathbb{R}$ on which $f$ is continuous
I'm very confused about how to think about this
real-analysis continuity uniform-continuity
edited Nov 16 at 5:24
Eevee Trainer
1,474216
asked Nov 16 at 5:08
kiarasaini
61
add a comment |
up vote
1
down vote
favorite
Consider the series
$$sum_{n=1}^infty frac{(-1)^n}{n+x}$$
Find all $x in mathbb{R}$ at which the series converges. Converges absolutely. Find all intervals of $mathbb{R}$ where the series defining $f$ converges uniformly, and all intervals of $mathbb{R}$ on which $f$ is continuous
I'm very confused about how to think about this
real-analysis continuity uniform-continuity
edited Nov 16 at 5:24
Eevee Trainer
1,474216
asked Nov 16 at 5:08
kiarasaini
61
you just need to know the definitions of conditional convergence, absolute convergence, uniform convergence and apply them to your series, it is not so complicate
– Masacroso
Nov 16 at 5:27
Do you know the conditions which guarantee the convergence of an alternating series?
– John Wayland Bales
Nov 16 at 5:28
@JohnWaylandBales it's not quite so simple if I'm not mistaken, at least for $x<0$.
– qbert
Nov 17 at 5:34
@qbert My question was intended to gauge just how confused the OP is. The first two should be easy aside from the negative integers.
– John Wayland Bales
Nov 17 at 6:27
@JohnWaylandBales fair enough and definitely true about the first two and uniform convergence on the nonegatives
– qbert
Nov 17 at 6:29
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Consider the series
$$sum_{n=1}^infty frac{(-1)^n}{n+x}$$
Find all $x in mathbb{R}$ at which the series converges. Converges absolutely. Find all intervals of $mathbb{R}$ where the series defining $f$ converges uniformly, and all intervals of $mathbb{R}$ on which $f$ is continuous
I'm very confused about how to think about this
real-analysis continuity uniform-continuity
edited Nov 16 at 5:24
Eevee Trainer
1,474216
asked Nov 16 at 5:08
kiarasaini
61
Consider the series
$$sum_{n=1}^infty frac{(-1)^n}{n+x}$$
Find all $x in mathbb{R}$ at which the series converges. Converges absolutely. Find all intervals of $mathbb{R}$ where the series defining $f$ converges uniformly, and all intervals of $mathbb{R}$ on which $f$ is continuous
I'm very confused about how to think about this
real-analysis continuity uniform-continuity
real-analysis continuity uniform-continuity
edited Nov 16 at 5:24
Eevee Trainer
1,474216
asked Nov 16 at 5:08
kiarasaini
61
edited Nov 16 at 5:24
Eevee Trainer
1,474216
asked Nov 16 at 5:08
kiarasaini
61
edited Nov 16 at 5:24
Eevee Trainer
1,474216
edited Nov 16 at 5:24
Eevee Trainer
1,474216
edited Nov 16 at 5:24
Eevee Trainer
1,474216
1,474216
asked Nov 16 at 5:08
kiarasaini
61
asked Nov 16 at 5:08
kiarasaini
61
asked Nov 16 at 5:08
kiarasaini
61
61
you just need to know the definitions of conditional convergence, absolute convergence, uniform convergence and apply them to your series, it is not so complicate
– Masacroso
Nov 16 at 5:27
Do you know the conditions which guarantee the convergence of an alternating series?
– John Wayland Bales
Nov 16 at 5:28
@JohnWaylandBales it's not quite so simple if I'm not mistaken, at least for $x<0$.
– qbert
Nov 17 at 5:34
@qbert My question was intended to gauge just how confused the OP is. The first two should be easy aside from the negative integers.
– John Wayland Bales
Nov 17 at 6:27
@JohnWaylandBales fair enough and definitely true about the first two and uniform convergence on the nonegatives
– qbert
Nov 17 at 6:29
add a comment |
you just need to know the definitions of conditional convergence, absolute convergence, uniform convergence and apply them to your series, it is not so complicate
– Masacroso
Nov 16 at 5:27
Do you know the conditions which guarantee the convergence of an alternating series?
– John Wayland Bales
Nov 16 at 5:28
@JohnWaylandBales it's not quite so simple if I'm not mistaken, at least for $x<0$.
– qbert
Nov 17 at 5:34
@qbert My question was intended to gauge just how confused the OP is. The first two should be easy aside from the negative integers.
– John Wayland Bales
Nov 17 at 6:27
@JohnWaylandBales fair enough and definitely true about the first two and uniform convergence on the nonegatives
– qbert
Nov 17 at 6:29
you just need to know the definitions of conditional convergence, absolute convergence, uniform convergence and apply them to your series, it is not so complicate
– Masacroso
Nov 16 at 5:27
you just need to know the definitions of conditional convergence, absolute convergence, uniform convergence and apply them to your series, it is not so complicate
– Masacroso
Nov 16 at 5:27
Do you know the conditions which guarantee the convergence of an alternating series?
– John Wayland Bales
Nov 16 at 5:28
Do you know the conditions which guarantee the convergence of an alternating series?
– John Wayland Bales
Nov 16 at 5:28
@JohnWaylandBales it's not quite so simple if I'm not mistaken, at least for $x<0$.
– qbert
Nov 17 at 5:34
@JohnWaylandBales it's not quite so simple if I'm not mistaken, at least for $x<0$.
– qbert
Nov 17 at 5:34
@qbert My question was intended to gauge just how confused the OP is. The first two should be easy aside from the negative integers.
– John Wayland Bales
Nov 17 at 6:27
@qbert My question was intended to gauge just how confused the OP is. The first two should be easy aside from the negative integers.
– John Wayland Bales
Nov 17 at 6:27
@JohnWaylandBales fair enough and definitely true about the first two and uniform convergence on the nonegatives
– qbert
Nov 17 at 6:29
@JohnWaylandBales fair enough and definitely true about the first two and uniform convergence on the nonegatives
– qbert
Nov 17 at 6:29
add a comment |
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you just need to know the definitions of conditional convergence, absolute convergence, uniform convergence and apply them to your series, it is not so complicate
– Masacroso
Nov 16 at 5:27
Do you know the conditions which guarantee the convergence of an alternating series?
– John Wayland Bales
Nov 16 at 5:28
@JohnWaylandBales it's not quite so simple if I'm not mistaken, at least for $x<0$.
– qbert
Nov 17 at 5:34
@qbert My question was intended to gauge just how confused the OP is. The first two should be easy aside from the negative integers.
– John Wayland Bales
Nov 17 at 6:27
@JohnWaylandBales fair enough and definitely true about the first two and uniform convergence on the nonegatives
– qbert
Nov 17 at 6:29