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Math Help - Urgent help with 2 VERY difficult problems please

  1. #1
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    Urgent help with 2 VERY difficult problems please

    For school, I need to solve the following two problems:
    1.) Prove, without calculator/computer, that 2010^2008<2009^2009<2008^2010.
    I believe using (n+1)^(n-1)<n^n<(n-1)^(n+1) then taking the ln of all three is the way to go.
    2.) Find the limit as n goes to infinity of 1/n*sqrt[(5n)!/(3n)!] the answer for this one is NOT 0.

    *Anyone that has the time and kindness PLEASE help me out. Thank you very much. And the answers do not require anything past Calculus II (integrals/limits etc)
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by eid03 View Post
    For school, I need to solve the following two problems:
    1.) Prove, without calculator/computer, that 2010^2008<2009^2009<2008^2010.
    I believe using (n+1)^(n-1)<n^n<(n-1)^(n+1) then taking the ln of all three is the way to go.
    2.) Find the limit as n goes to infinity of 1/n*sqrt[(5n)!/(3n)!] the answer for this one is NOT 0.

    *Anyone that has the time and kindness PLEASE help me out. Thank you very much. And the answers do not require anything past Calculus II (integrals/limits etc)
    The first one is actually kind of easy. Show some actual work and we'll help (it has a clever solution).

    I don't understand what the second one is? It's either infinity or zero. Th syntax is unreadable.
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  3. #3
    MHF Contributor Drexel28's Avatar
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    The above (LHS) is equivalent to proving that 2008\ln(2010)<2009\ln(2009). Recall for a second that f\left(x+\Delta x\right)\approx \Delta xf'(x)+f(x) and in this case ew have that \ln\left(x+\Delta x\right)< \frac{\Delta x}{x}+f(x). Taking \Delta x=1,x=2009 wee see that 2008\ln(2010)<\frac{2008}{2009}+2008\ln(2009) where the conclusion readily follows.
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