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MATH 2432 PROJECT # 3 NAME :__________ 1. (Mathematica ) Choose the value of "m" , and plot the curve given by : r(t) a sin(mt) for various...

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MATH 2432 PROJECT # 3 NAME :__________ 1. ( Mathematica ) Choose the value of "m" , and plot the curve given by : r(t) a sin(mt) for various positive values of "a", such as 0.2, 0.4, 0.6, etc. See what happens as the number a decreases toward zero. Try this for both even and odd values of "m". What do you learn about the difference between sin(mt) for m even and m odd ? What does r sin(mt) look like when m is not an integer ? In this case be sure to choose an interval for t larger than [0 ,2 ]. How large should this interval be ? 2. As n increases, there is very little change in the difference between the sum 1 1 2 1 3 ... 1 n and the integral ln n 1 n 1 x dx To explore the idea, carry out the following steps : a) By taking f(x) 1/ x in the double inequality that is used in the proof of the integral test, show that ln(n 1) 1 1 2 1 3 ... 1 n ln n or 0 ln(n 1) ln n 1 1 2 1 3 ... 1 n ln n 1. Thus, the sequence a n 1 1 2 1 3 ... 1 n ln n is bounded from below and from above. b) Show that 1 n 1 n n 1 1 x dx ln(n 1) ln n and then show that the sequence {a n } in part (a) is decreasing. Since a decreasing sequence that is bounded from below converges, the numbers a n defined in (a) converge :
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1 1 2 1 3 ... 1 n ln n . Note : The number , whose value is 0.5772. .. , is called Euler’s constant. In contrast to other numbers like or e, no other expression with a simple law of formulation has ever been found for .
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