Re: Convergence/Divergence
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I am reviewing for a math exam and am having some trouble with sequences and convergence/divergence.
One of the practice problems is:
1) Determine if the sequence {[2^(2n)]/[3^(n+2)]} from n=1 to infinity is convergent or divergent.
I am having a difficult time with this problem because of the exponents. If anyone could show me how to do this that would be great.
The second problem is:
2) Use the squeeze theorem to show that the sequence {[1x3x5x7...x(2n-3)x(2n-1)]/[(2n)^n] from n=1 to infinity is convergent to 0.
I really don't understand how to use the squeeze theorem here. Again, any help would be great.
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In #1, you probably want to manipulate it a bit to make it easier to work with. In other words, make it so both the numerator and denominator have the same exponent. I would start by factoring out 3^2 from the denominator. So lim ([2^(2n)]/[3^(n+2)] = lim (1/9)*[2^(2n)]/[3^n]. Now notice that power of 2n is going to be difficult to work with, so instead consider 2^(2n) as (2^2)^n=4^n. Thus you have lim (1/9)*[(4^n)/(3^n)] = lim (1/9)*(4/3)^n. Since 4/3>1, this diverges towards infinity. Of course you probably shouldn't strictly trust the results of some random message board poster that you don't know...
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