Results 1 to 3 of 3

Thread: [SOLVED] Calculating limit of a sequence

  1. #1
    Senior Member mollymcf2009's Avatar
    Joined
    Jan 2009
    From
    Charleston, SC
    Posts
    490
    Awards
    1

    [SOLVED] Calculating limit of a sequence

    Can someone help me with a good strategy for finding the limit of a sequence? I am just not understanding the best way to solve these. What I have been trying to do to first take the limit. If i got an indeterminate, I changed it to a function and took the derivative, but I'm not sure that is the way I need to evaluate all of these. Any help is greatly appreciated!

    1) $\displaystyle a_n = 1 - (0.1)^n
    $

    $\displaystyle \lim_{n\rightarrow \infty}$ $\displaystyle a_n = ?$


    2) $\displaystyle a_n = \frac{n^3}{3n + 1}$

    $\displaystyle \lim_{n\rightarrow \infty}$ $\displaystyle a_n = ?$


    3) $\displaystyle a_n = e^{7/(n+9)}$

    $\displaystyle
    \lim_{n\rightarrow \infty}$ $\displaystyle a_n = ?$


    4) $\displaystyle a_n = \frac{3^{n+1}}{5^n}$

    $\displaystyle \lim_{n\rightarrow \infty}$ $\displaystyle a_n = ?$


    5) $\displaystyle a_n = tan (\frac{6n\pi}{4 + 24n})$

    $\displaystyle \lim_{n\rightarrow \infty}$ $\displaystyle a_n = ?$


    6) $\displaystyle a_n = n^2e^{-5n}$

    $\displaystyle \lim_{n\rightarrow \infty}$ $\displaystyle a_n = ?$
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Rhymes with Orange Chris L T521's Avatar
    Joined
    May 2008
    From
    Chicago, IL
    Posts
    2,844
    Thanks
    5
    Quote Originally Posted by mollymcf2009 View Post
    Can someone help me with a good strategy for finding the limit of a sequence? I am just not understanding the best way to solve these. What I have been trying to do to first take the limit. If i got an indeterminate, I changed it to a function and took the derivative, but I'm not sure that is the way I need to evaluate all of these. Any help is greatly appreciated!

    1) $\displaystyle a_n = 1 - (0.1)^n
    $

    $\displaystyle \lim_{n\rightarrow \infty}$ $\displaystyle a_n = ?$
    Hint: $\displaystyle \lim a^n=0$ when $\displaystyle \left|a\right|<1$

    2) $\displaystyle a_n = \frac{n^3}{3n + 1}$

    $\displaystyle \lim_{n\rightarrow \infty}$ $\displaystyle a_n = ?$
    Hint: Divide the numerator and denominator through by $\displaystyle n^3$ to get $\displaystyle \lim\frac{1}{\frac{3}{n^2}+\frac{1}{n^3}}$

    Can you take it from here?

    3) $\displaystyle a_n = e^{7/(n+9)}$

    $\displaystyle
    \lim_{n\rightarrow \infty}$ $\displaystyle a_n = ?$
    $\displaystyle \lim e^{\frac{7}{n+9}}=e^{\lim\frac{7}{n+9}}$

    Can you take it from here?

    4) $\displaystyle a_n = \frac{3^{n+1}}{5^n}$

    $\displaystyle \lim_{n\rightarrow \infty}$ $\displaystyle a_n = ?$
    $\displaystyle \lim\frac{3^{n+1}}{5^n}=3\lim\frac{3^n}{5^n}=3\lim \left(\frac{3}{5}\right)^n$. Now see my hint for #1


    5) $\displaystyle a_n = tan (\frac{6n\pi}{4 + 24n})$

    $\displaystyle \lim_{n\rightarrow \infty}$ $\displaystyle a_n = ?$
    $\displaystyle \lim\tan\left(\frac{6n\pi}{24n+4}\right)=\tan\left (\lim\frac{6\pi n}{24n+4}\right)$

    Can you take it from here?

    6) $\displaystyle a_n = n^2e^{-5n}$

    $\displaystyle \lim_{n\rightarrow \infty}$ $\displaystyle a_n = ?$
    $\displaystyle \lim n^2e^{-5n}=\lim\frac{n^2}{e^{5n}}$

    Can you take it from here?



    (Note that $\displaystyle \lim$ is analogous with $\displaystyle \lim_{n\to\infty}$)
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member mollymcf2009's Avatar
    Joined
    Jan 2009
    From
    Charleston, SC
    Posts
    490
    Awards
    1
    Quote Originally Posted by mollymcf2009 View Post
    can someone help me with a good strategy for finding the limit of a sequence? I am just not understanding the best way to solve these. What i have been trying to do to first take the limit. If i got an indeterminate, i changed it to a function and took the derivative, but i'm not sure that is the way i need to evaluate all of these. Any help is greatly appreciated!

    1) $\displaystyle a_n = 1 - (0.1)^n
    $

    $\displaystyle \lim_{n\rightarrow \infty}$ $\displaystyle a_n = ?$ 1


    2) $\displaystyle a_n = \frac{n^3}{3n + 1}$

    $\displaystyle \lim_{n\rightarrow \infty}$ $\displaystyle a_n = ?$ diverges


    3) $\displaystyle a_n = e^{7/(n+9)}$

    $\displaystyle
    \lim_{n\rightarrow \infty}$ $\displaystyle a_n = ?$ 1


    4) $\displaystyle a_n = \frac{3^{n+1}}{5^n}$

    $\displaystyle \lim_{n\rightarrow \infty}$ $\displaystyle a_n = ?$ 0


    5) $\displaystyle a_n = tan (\frac{6n\pi}{4 + 24n})$

    $\displaystyle \lim_{n\rightarrow \infty}$ $\displaystyle a_n = ?$ 1


    6) $\displaystyle a_n = n^2e^{-5n}$

    $\displaystyle \lim_{n\rightarrow \infty}$ $\displaystyle a_n = ?$ 0


    correct answers are in red above for future reference
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. calculating a limit
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Apr 10th 2011, 05:37 PM
  2. Calculating a Limit
    Posted in the Calculus Forum
    Replies: 8
    Last Post: Nov 1st 2010, 02:16 PM
  3. Need help Calculating the limit?
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Oct 4th 2010, 04:18 PM
  4. [SOLVED] Limit of a sequence - recursion
    Posted in the Calculus Forum
    Replies: 0
    Last Post: Dec 4th 2009, 12:54 AM
  5. [SOLVED] Finding the limit of a sequence.
    Posted in the Calculus Forum
    Replies: 5
    Last Post: Nov 15th 2008, 06:38 AM

Search Tags


/mathhelpforum @mathhelpforum