$lim_{ntoinfty} n^{x}(a_{1}a_{2}ldots a_{n})^{frac{1}{n}}=ae^x$
up vote
2
down vote
favorite
Let $ (a_{n})$ be positive sequence, $a,x in R quad $ and $ lim_{ntoinfty} n^{x}a_{n}=a$.
Prove that $lim_{ntoinfty} n^{x}(a_{1}a_{2}ldots a_{n})^{frac{1}{n}}=ae^x$
I know that $lim_{ntoinfty} (a_{1}a_{2}ldots a_{n})^{frac{1}{n}}=lim_{ntoinfty} a_{n}$ but don't have idea how to use it
limits
add a comment |
up vote
2
down vote
favorite
Let $ (a_{n})$ be positive sequence, $a,x in R quad $ and $ lim_{ntoinfty} n^{x}a_{n}=a$.
Prove that $lim_{ntoinfty} n^{x}(a_{1}a_{2}ldots a_{n})^{frac{1}{n}}=ae^x$
I know that $lim_{ntoinfty} (a_{1}a_{2}ldots a_{n})^{frac{1}{n}}=lim_{ntoinfty} a_{n}$ but don't have idea how to use it
limits
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Let $ (a_{n})$ be positive sequence, $a,x in R quad $ and $ lim_{ntoinfty} n^{x}a_{n}=a$.
Prove that $lim_{ntoinfty} n^{x}(a_{1}a_{2}ldots a_{n})^{frac{1}{n}}=ae^x$
I know that $lim_{ntoinfty} (a_{1}a_{2}ldots a_{n})^{frac{1}{n}}=lim_{ntoinfty} a_{n}$ but don't have idea how to use it
limits
Let $ (a_{n})$ be positive sequence, $a,x in R quad $ and $ lim_{ntoinfty} n^{x}a_{n}=a$.
Prove that $lim_{ntoinfty} n^{x}(a_{1}a_{2}ldots a_{n})^{frac{1}{n}}=ae^x$
I know that $lim_{ntoinfty} (a_{1}a_{2}ldots a_{n})^{frac{1}{n}}=lim_{ntoinfty} a_{n}$ but don't have idea how to use it
limits
limits
edited Nov 20 at 22:45
gimusi
91k74495
91k74495
asked Nov 20 at 22:29
math.trouble
496
496
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
up vote
3
down vote
accepted
HINT
By ratio root criteria we have
$$frac{(n+1)^{x(n+1)}a_{1}a_{2}ldots a_{n+1}}{n^{xn}a_{1}a_{2}ldots a_{n}}=(n+1)^xa_{n+1}left(1+frac1nright)^{nx}$$
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
accepted
HINT
By ratio root criteria we have
$$frac{(n+1)^{x(n+1)}a_{1}a_{2}ldots a_{n+1}}{n^{xn}a_{1}a_{2}ldots a_{n}}=(n+1)^xa_{n+1}left(1+frac1nright)^{nx}$$
add a comment |
up vote
3
down vote
accepted
HINT
By ratio root criteria we have
$$frac{(n+1)^{x(n+1)}a_{1}a_{2}ldots a_{n+1}}{n^{xn}a_{1}a_{2}ldots a_{n}}=(n+1)^xa_{n+1}left(1+frac1nright)^{nx}$$
add a comment |
up vote
3
down vote
accepted
up vote
3
down vote
accepted
HINT
By ratio root criteria we have
$$frac{(n+1)^{x(n+1)}a_{1}a_{2}ldots a_{n+1}}{n^{xn}a_{1}a_{2}ldots a_{n}}=(n+1)^xa_{n+1}left(1+frac1nright)^{nx}$$
HINT
By ratio root criteria we have
$$frac{(n+1)^{x(n+1)}a_{1}a_{2}ldots a_{n+1}}{n^{xn}a_{1}a_{2}ldots a_{n}}=(n+1)^xa_{n+1}left(1+frac1nright)^{nx}$$
answered Nov 20 at 22:43
gimusi
91k74495
91k74495
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3006994%2flim-n-to-infty-nxa-1a-2-ldots-a-n-frac1n-aex%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown