Mixing
Pointwise convergence of $sum_{n=0}^infty frac{1}{2^nsqrt{1+nx}}$
$begingroup$
Given this series :
$$
sum_{n=0}^infty frac{1}{2^n*sqrt{1+nx}}
$$
I have to prove for which $x geq 0$ the series converges pointwise.
if $x=0$ the series is :
$$sum_{n=0}^infty Big(frac{1}{2}Big)^n$$
and this series converges.
If I assume that $xneq 0$ I don't find the convergence.
How can I proceed? Do I have to consider a majorant series?
Thanks in advance for any help.
real-analysis convergence pointwise-convergence
edited Jan 22 at 19:11
Martin Sleziak
44.8k10119273
asked Jan 22 at 17:19
andrewandrew
698
$endgroup$
add a comment |
$begingroup$
Given this series :
$$
sum_{n=0}^infty frac{1}{2^n*sqrt{1+nx}}
$$
I have to prove for which $x geq 0$ the series converges pointwise.
if $x=0$ the series is :
$$sum_{n=0}^infty Big(frac{1}{2}Big)^n$$
and this series converges.
If I assume that $xneq 0$ I don't find the convergence.
How can I proceed? Do I have to consider a majorant series?
Thanks in advance for any help.
real-analysis convergence pointwise-convergence
edited Jan 22 at 19:11
Martin Sleziak
44.8k10119273
asked Jan 22 at 17:19
andrewandrew
698
$endgroup$
5
$begingroup$
Do I have to consider a majorant series? YES. this series is smaller than $$sum_n frac{1}{2^n}$$ which is convergent, thus it converges for all $x ge 0$.
$endgroup$
– Crostul
Jan 22 at 17:20
1
$begingroup$
ok you're right @Crostul
$endgroup$
– andrew
Jan 22 at 17:24
add a comment |
$begingroup$
Given this series :
$$
sum_{n=0}^infty frac{1}{2^n*sqrt{1+nx}}
$$
I have to prove for which $x geq 0$ the series converges pointwise.
if $x=0$ the series is :
$$sum_{n=0}^infty Big(frac{1}{2}Big)^n$$
and this series converges.
If I assume that $xneq 0$ I don't find the convergence.
How can I proceed? Do I have to consider a majorant series?
Thanks in advance for any help.
real-analysis convergence pointwise-convergence
edited Jan 22 at 19:11
Martin Sleziak
44.8k10119273
asked Jan 22 at 17:19
andrewandrew
698
$endgroup$
Given this series :
$$
sum_{n=0}^infty frac{1}{2^n*sqrt{1+nx}}
$$
I have to prove for which $x geq 0$ the series converges pointwise.
if $x=0$ the series is :
$$sum_{n=0}^infty Big(frac{1}{2}Big)^n$$
and this series converges.
If I assume that $xneq 0$ I don't find the convergence.
How can I proceed? Do I have to consider a majorant series?
Thanks in advance for any help.
real-analysis convergence pointwise-convergence
real-analysis convergence pointwise-convergence
edited Jan 22 at 19:11
Martin Sleziak
44.8k10119273
asked Jan 22 at 17:19
andrewandrew
698
edited Jan 22 at 19:11
Martin Sleziak
44.8k10119273
asked Jan 22 at 17:19
andrewandrew
698
edited Jan 22 at 19:11
Martin Sleziak
44.8k10119273
edited Jan 22 at 19:11
Martin Sleziak
44.8k10119273
edited Jan 22 at 19:11
Martin Sleziak
44.8k10119273
44.8k10119273
asked Jan 22 at 17:19
andrewandrew
698
asked Jan 22 at 17:19
andrewandrew
698
asked Jan 22 at 17:19
andrewandrew
698
698
5
$begingroup$
Do I have to consider a majorant series? YES. this series is smaller than $$sum_n frac{1}{2^n}$$ which is convergent, thus it converges for all $x ge 0$.
$endgroup$
– Crostul
Jan 22 at 17:20
1
$begingroup$
ok you're right @Crostul
$endgroup$
– andrew
Jan 22 at 17:24
add a comment |
5
$begingroup$
Do I have to consider a majorant series? YES. this series is smaller than $$sum_n frac{1}{2^n}$$ which is convergent, thus it converges for all $x ge 0$.
$endgroup$
– Crostul
Jan 22 at 17:20
1
$begingroup$
ok you're right @Crostul
$endgroup$
– andrew
Jan 22 at 17:24
5
5
$begingroup$
Do I have to consider a majorant series? YES. this series is smaller than $$sum_n frac{1}{2^n}$$ which is convergent, thus it converges for all $x ge 0$.
$endgroup$
– Crostul
Jan 22 at 17:20
$begingroup$
Do I have to consider a majorant series? YES. this series is smaller than $$sum_n frac{1}{2^n}$$ which is convergent, thus it converges for all $x ge 0$.
$endgroup$
– Crostul
Jan 22 at 17:20
1
1
$begingroup$
ok you're right @Crostul
$endgroup$
– andrew
Jan 22 at 17:24
$begingroup$
ok you're right @Crostul
$endgroup$
– andrew
Jan 22 at 17:24
add a comment |
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$begingroup$
Do I have to consider a majorant series? YES. this series is smaller than $$sum_n frac{1}{2^n}$$ which is convergent, thus it converges for all $x ge 0$.
$endgroup$
– Crostul
Jan 22 at 17:20
1
$begingroup$
ok you're right @Crostul
$endgroup$
– andrew
Jan 22 at 17:24