How to evaluate $ lim_{nto infty} frac{i}{n}left(frac{1+i}{sqrt{2}}right)^n $ where $i=sqrt{-1}$?
$begingroup$
How to evaluate the following limit?
$$
lim_{nto infty} frac{i}{n}left(frac{1+i}{sqrt{2}}right)^n
$$
Here $i=sqrt{-1}$.
I got:
$$lim_{nto infty} frac{i}{n}left(frac{1+i}{sqrt{2}}right)^n
= lim_{nto infty} frac{(i-1)^n}{n(sqrt{2})^n}
$$
I know the lower part goes to infinity but what to do with the upper part? Is that usefull to use squeeze theorem or is there any simplier way?
sequences-and-series limits complex-numbers
$endgroup$
|
show 2 more comments
$begingroup$
How to evaluate the following limit?
$$
lim_{nto infty} frac{i}{n}left(frac{1+i}{sqrt{2}}right)^n
$$
Here $i=sqrt{-1}$.
I got:
$$lim_{nto infty} frac{i}{n}left(frac{1+i}{sqrt{2}}right)^n
= lim_{nto infty} frac{(i-1)^n}{n(sqrt{2})^n}
$$
I know the lower part goes to infinity but what to do with the upper part? Is that usefull to use squeeze theorem or is there any simplier way?
sequences-and-series limits complex-numbers
$endgroup$
1
$begingroup$
It's very hard to understand what you wrote. Try mathjax
$endgroup$
– DonAntonio
Nov 24 '18 at 21:38
$begingroup$
Express $(1+i)/sqrt{2}=exp(ipi/4)$.
$endgroup$
– Diger
Nov 24 '18 at 21:39
1
$begingroup$
L.Spy Talk of "mathjax" might be confusing to you. Sorry about that. Mathjax is a way to format mathematical expressions so they render very nicely, like you'd see in a textbook. Here is a really handy tutorial for learning mathjax. Anything you want under a square root sign, you can format as$sqrt{blah blah}$
. A limit of a function f(x) from $n to infty$ can be written$lim_{nto infty} f(x)$
. Just a few pointers.
$endgroup$
– amWhy
Nov 24 '18 at 21:43
1
$begingroup$
If your function $f(x)$ is a fraction, you can write it as follows$f(x) = frac{"numerator here"}{"denominator here"}$
$endgroup$
– amWhy
Nov 24 '18 at 21:46
$begingroup$
Write $frac{1+i}{sqrt{2}}=e^{ipi/4}implies (frac{1+i}{sqrt{2}})^n=e^{i npi/4}$
$endgroup$
– Shubham Johri
Nov 24 '18 at 21:57
|
show 2 more comments
$begingroup$
How to evaluate the following limit?
$$
lim_{nto infty} frac{i}{n}left(frac{1+i}{sqrt{2}}right)^n
$$
Here $i=sqrt{-1}$.
I got:
$$lim_{nto infty} frac{i}{n}left(frac{1+i}{sqrt{2}}right)^n
= lim_{nto infty} frac{(i-1)^n}{n(sqrt{2})^n}
$$
I know the lower part goes to infinity but what to do with the upper part? Is that usefull to use squeeze theorem or is there any simplier way?
sequences-and-series limits complex-numbers
$endgroup$
How to evaluate the following limit?
$$
lim_{nto infty} frac{i}{n}left(frac{1+i}{sqrt{2}}right)^n
$$
Here $i=sqrt{-1}$.
I got:
$$lim_{nto infty} frac{i}{n}left(frac{1+i}{sqrt{2}}right)^n
= lim_{nto infty} frac{(i-1)^n}{n(sqrt{2})^n}
$$
I know the lower part goes to infinity but what to do with the upper part? Is that usefull to use squeeze theorem or is there any simplier way?
sequences-and-series limits complex-numbers
sequences-and-series limits complex-numbers
edited Nov 24 '18 at 23:16
user587192
1,789315
1,789315
asked Nov 24 '18 at 21:35
L. SpyL. Spy
12
12
1
$begingroup$
It's very hard to understand what you wrote. Try mathjax
$endgroup$
– DonAntonio
Nov 24 '18 at 21:38
$begingroup$
Express $(1+i)/sqrt{2}=exp(ipi/4)$.
$endgroup$
– Diger
Nov 24 '18 at 21:39
1
$begingroup$
L.Spy Talk of "mathjax" might be confusing to you. Sorry about that. Mathjax is a way to format mathematical expressions so they render very nicely, like you'd see in a textbook. Here is a really handy tutorial for learning mathjax. Anything you want under a square root sign, you can format as$sqrt{blah blah}$
. A limit of a function f(x) from $n to infty$ can be written$lim_{nto infty} f(x)$
. Just a few pointers.
$endgroup$
– amWhy
Nov 24 '18 at 21:43
1
$begingroup$
If your function $f(x)$ is a fraction, you can write it as follows$f(x) = frac{"numerator here"}{"denominator here"}$
$endgroup$
– amWhy
Nov 24 '18 at 21:46
$begingroup$
Write $frac{1+i}{sqrt{2}}=e^{ipi/4}implies (frac{1+i}{sqrt{2}})^n=e^{i npi/4}$
$endgroup$
– Shubham Johri
Nov 24 '18 at 21:57
|
show 2 more comments
1
$begingroup$
It's very hard to understand what you wrote. Try mathjax
$endgroup$
– DonAntonio
Nov 24 '18 at 21:38
$begingroup$
Express $(1+i)/sqrt{2}=exp(ipi/4)$.
$endgroup$
– Diger
Nov 24 '18 at 21:39
1
$begingroup$
L.Spy Talk of "mathjax" might be confusing to you. Sorry about that. Mathjax is a way to format mathematical expressions so they render very nicely, like you'd see in a textbook. Here is a really handy tutorial for learning mathjax. Anything you want under a square root sign, you can format as$sqrt{blah blah}$
. A limit of a function f(x) from $n to infty$ can be written$lim_{nto infty} f(x)$
. Just a few pointers.
$endgroup$
– amWhy
Nov 24 '18 at 21:43
1
$begingroup$
If your function $f(x)$ is a fraction, you can write it as follows$f(x) = frac{"numerator here"}{"denominator here"}$
$endgroup$
– amWhy
Nov 24 '18 at 21:46
$begingroup$
Write $frac{1+i}{sqrt{2}}=e^{ipi/4}implies (frac{1+i}{sqrt{2}})^n=e^{i npi/4}$
$endgroup$
– Shubham Johri
Nov 24 '18 at 21:57
1
1
$begingroup$
It's very hard to understand what you wrote. Try mathjax
$endgroup$
– DonAntonio
Nov 24 '18 at 21:38
$begingroup$
It's very hard to understand what you wrote. Try mathjax
$endgroup$
– DonAntonio
Nov 24 '18 at 21:38
$begingroup$
Express $(1+i)/sqrt{2}=exp(ipi/4)$.
$endgroup$
– Diger
Nov 24 '18 at 21:39
$begingroup$
Express $(1+i)/sqrt{2}=exp(ipi/4)$.
$endgroup$
– Diger
Nov 24 '18 at 21:39
1
1
$begingroup$
L.Spy Talk of "mathjax" might be confusing to you. Sorry about that. Mathjax is a way to format mathematical expressions so they render very nicely, like you'd see in a textbook. Here is a really handy tutorial for learning mathjax. Anything you want under a square root sign, you can format as
$sqrt{blah blah}$
. A limit of a function f(x) from $n to infty$ can be written $lim_{nto infty} f(x)$
. Just a few pointers.$endgroup$
– amWhy
Nov 24 '18 at 21:43
$begingroup$
L.Spy Talk of "mathjax" might be confusing to you. Sorry about that. Mathjax is a way to format mathematical expressions so they render very nicely, like you'd see in a textbook. Here is a really handy tutorial for learning mathjax. Anything you want under a square root sign, you can format as
$sqrt{blah blah}$
. A limit of a function f(x) from $n to infty$ can be written $lim_{nto infty} f(x)$
. Just a few pointers.$endgroup$
– amWhy
Nov 24 '18 at 21:43
1
1
$begingroup$
If your function $f(x)$ is a fraction, you can write it as follows
$f(x) = frac{"numerator here"}{"denominator here"}$
$endgroup$
– amWhy
Nov 24 '18 at 21:46
$begingroup$
If your function $f(x)$ is a fraction, you can write it as follows
$f(x) = frac{"numerator here"}{"denominator here"}$
$endgroup$
– amWhy
Nov 24 '18 at 21:46
$begingroup$
Write $frac{1+i}{sqrt{2}}=e^{ipi/4}implies (frac{1+i}{sqrt{2}})^n=e^{i npi/4}$
$endgroup$
– Shubham Johri
Nov 24 '18 at 21:57
$begingroup$
Write $frac{1+i}{sqrt{2}}=e^{ipi/4}implies (frac{1+i}{sqrt{2}})^n=e^{i npi/4}$
$endgroup$
– Shubham Johri
Nov 24 '18 at 21:57
|
show 2 more comments
2 Answers
2
active
oldest
votes
$begingroup$
We have
$$
left|frac{1+i}{sqrt{2}}right|=1
$$
Thus
$$
left|frac{i}{n}left(frac{1+i}{sqrt{2}}right)^{!n}right|=frac{1}{n}
$$
$endgroup$
$begingroup$
egreg-How it could be 1? I thought that (1+i)/√2 is one of the roots of i=(a+bi)^2
$endgroup$
– L. Spy
Nov 25 '18 at 9:25
$begingroup$
@L.Spy $sqrt{(1/sqrt{2})^2+(1/sqrt{2})^2}=sqrt{1/2+1/2}=1$. With your (more complicated) approach: since $|i|=1$, also its square roots have modulus $1$.
$endgroup$
– egreg
Nov 25 '18 at 9:40
add a comment |
$begingroup$
$$lim_{nto infty} |frac{i}{n}left(frac{1+i}{sqrt{2}}right)^n|=lim_{nto infty} {1over n}=0$$therefore $$lim_{nto infty} frac{i}{n}left(frac{1+i}{sqrt{2}}right)^n=0$$
$endgroup$
add a comment |
Your Answer
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
We have
$$
left|frac{1+i}{sqrt{2}}right|=1
$$
Thus
$$
left|frac{i}{n}left(frac{1+i}{sqrt{2}}right)^{!n}right|=frac{1}{n}
$$
$endgroup$
$begingroup$
egreg-How it could be 1? I thought that (1+i)/√2 is one of the roots of i=(a+bi)^2
$endgroup$
– L. Spy
Nov 25 '18 at 9:25
$begingroup$
@L.Spy $sqrt{(1/sqrt{2})^2+(1/sqrt{2})^2}=sqrt{1/2+1/2}=1$. With your (more complicated) approach: since $|i|=1$, also its square roots have modulus $1$.
$endgroup$
– egreg
Nov 25 '18 at 9:40
add a comment |
$begingroup$
We have
$$
left|frac{1+i}{sqrt{2}}right|=1
$$
Thus
$$
left|frac{i}{n}left(frac{1+i}{sqrt{2}}right)^{!n}right|=frac{1}{n}
$$
$endgroup$
$begingroup$
egreg-How it could be 1? I thought that (1+i)/√2 is one of the roots of i=(a+bi)^2
$endgroup$
– L. Spy
Nov 25 '18 at 9:25
$begingroup$
@L.Spy $sqrt{(1/sqrt{2})^2+(1/sqrt{2})^2}=sqrt{1/2+1/2}=1$. With your (more complicated) approach: since $|i|=1$, also its square roots have modulus $1$.
$endgroup$
– egreg
Nov 25 '18 at 9:40
add a comment |
$begingroup$
We have
$$
left|frac{1+i}{sqrt{2}}right|=1
$$
Thus
$$
left|frac{i}{n}left(frac{1+i}{sqrt{2}}right)^{!n}right|=frac{1}{n}
$$
$endgroup$
We have
$$
left|frac{1+i}{sqrt{2}}right|=1
$$
Thus
$$
left|frac{i}{n}left(frac{1+i}{sqrt{2}}right)^{!n}right|=frac{1}{n}
$$
answered Nov 24 '18 at 22:53
egregegreg
180k1485202
180k1485202
$begingroup$
egreg-How it could be 1? I thought that (1+i)/√2 is one of the roots of i=(a+bi)^2
$endgroup$
– L. Spy
Nov 25 '18 at 9:25
$begingroup$
@L.Spy $sqrt{(1/sqrt{2})^2+(1/sqrt{2})^2}=sqrt{1/2+1/2}=1$. With your (more complicated) approach: since $|i|=1$, also its square roots have modulus $1$.
$endgroup$
– egreg
Nov 25 '18 at 9:40
add a comment |
$begingroup$
egreg-How it could be 1? I thought that (1+i)/√2 is one of the roots of i=(a+bi)^2
$endgroup$
– L. Spy
Nov 25 '18 at 9:25
$begingroup$
@L.Spy $sqrt{(1/sqrt{2})^2+(1/sqrt{2})^2}=sqrt{1/2+1/2}=1$. With your (more complicated) approach: since $|i|=1$, also its square roots have modulus $1$.
$endgroup$
– egreg
Nov 25 '18 at 9:40
$begingroup$
egreg-How it could be 1? I thought that (1+i)/√2 is one of the roots of i=(a+bi)^2
$endgroup$
– L. Spy
Nov 25 '18 at 9:25
$begingroup$
egreg-How it could be 1? I thought that (1+i)/√2 is one of the roots of i=(a+bi)^2
$endgroup$
– L. Spy
Nov 25 '18 at 9:25
$begingroup$
@L.Spy $sqrt{(1/sqrt{2})^2+(1/sqrt{2})^2}=sqrt{1/2+1/2}=1$. With your (more complicated) approach: since $|i|=1$, also its square roots have modulus $1$.
$endgroup$
– egreg
Nov 25 '18 at 9:40
$begingroup$
@L.Spy $sqrt{(1/sqrt{2})^2+(1/sqrt{2})^2}=sqrt{1/2+1/2}=1$. With your (more complicated) approach: since $|i|=1$, also its square roots have modulus $1$.
$endgroup$
– egreg
Nov 25 '18 at 9:40
add a comment |
$begingroup$
$$lim_{nto infty} |frac{i}{n}left(frac{1+i}{sqrt{2}}right)^n|=lim_{nto infty} {1over n}=0$$therefore $$lim_{nto infty} frac{i}{n}left(frac{1+i}{sqrt{2}}right)^n=0$$
$endgroup$
add a comment |
$begingroup$
$$lim_{nto infty} |frac{i}{n}left(frac{1+i}{sqrt{2}}right)^n|=lim_{nto infty} {1over n}=0$$therefore $$lim_{nto infty} frac{i}{n}left(frac{1+i}{sqrt{2}}right)^n=0$$
$endgroup$
add a comment |
$begingroup$
$$lim_{nto infty} |frac{i}{n}left(frac{1+i}{sqrt{2}}right)^n|=lim_{nto infty} {1over n}=0$$therefore $$lim_{nto infty} frac{i}{n}left(frac{1+i}{sqrt{2}}right)^n=0$$
$endgroup$
$$lim_{nto infty} |frac{i}{n}left(frac{1+i}{sqrt{2}}right)^n|=lim_{nto infty} {1over n}=0$$therefore $$lim_{nto infty} frac{i}{n}left(frac{1+i}{sqrt{2}}right)^n=0$$
answered Nov 28 '18 at 0:31
Mostafa AyazMostafa Ayaz
15.2k3939
15.2k3939
add a comment |
add a comment |
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1
$begingroup$
It's very hard to understand what you wrote. Try mathjax
$endgroup$
– DonAntonio
Nov 24 '18 at 21:38
$begingroup$
Express $(1+i)/sqrt{2}=exp(ipi/4)$.
$endgroup$
– Diger
Nov 24 '18 at 21:39
1
$begingroup$
L.Spy Talk of "mathjax" might be confusing to you. Sorry about that. Mathjax is a way to format mathematical expressions so they render very nicely, like you'd see in a textbook. Here is a really handy tutorial for learning mathjax. Anything you want under a square root sign, you can format as
$sqrt{blah blah}$
. A limit of a function f(x) from $n to infty$ can be written$lim_{nto infty} f(x)$
. Just a few pointers.$endgroup$
– amWhy
Nov 24 '18 at 21:43
1
$begingroup$
If your function $f(x)$ is a fraction, you can write it as follows
$f(x) = frac{"numerator here"}{"denominator here"}$
$endgroup$
– amWhy
Nov 24 '18 at 21:46
$begingroup$
Write $frac{1+i}{sqrt{2}}=e^{ipi/4}implies (frac{1+i}{sqrt{2}})^n=e^{i npi/4}$
$endgroup$
– Shubham Johri
Nov 24 '18 at 21:57