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## Exercises.

1. For each of the series A-D below do the following.

a.)Use a suitable test to determine if the series is convergent or divergent. (You can, but don't have to use Maple for this.)

b.)Plot a sufficient number (20-50) of partial sums of the series to illustrate its convergence or divergence.

c.)If the series is convergent, find its sum.

d.)If the series is convergent, determine the number of terms that needed to be added up, in order to approximate the sum of the series with an error not greater than 0.001.

A.

B.

C.

D.

2.We will use each of the following three closely interrelated series to approximate . Note that is an irrational number which can only be approximated but never exactly expressed in terms of finite decimals.

Write out a few terms of the series to discover how they are connected to each other.

a.)Find the sums of all three series.

b.)Since all three limits involve the partial sums of all three series could be used to approximate this irrational number. Plot the partial sums of the three series that will converge to . Use the plot to determine the minimum number of terms that need to be added up to approximate the value of with an error not greater than 0.01.

c.)For  Ser3 use the alternating series error estimate to answer 2.b. Explain why the number of necessary terms is not the same as what you got by the graphical inspection in 2.b. Discuss the comparative advantages and disadvantages of the graphical method vs. the error formula.

3.(EXTRA CREDIT - 8 points) In this problem you will discover that the difference between a nuclear bomb and a power station is just the convergence or divergence of an infinite series.

The simplified model of a reaction is as follows. Particles are flying, and occasionally colliding, in a closed reactor vessel. At every collision a new particle is born, increasing the total number of particles by one, and also a fixed amount of energy is freed. The energy released at each collision increases the speed of the particles. According to the kinetical energy formula

the average speed of the particles after the n-th collision will be proportional to .

As time passes, more and more particles will be flying in the reactor faster and faster. Consequently collisions will occur with rapidly increasing frequency. The time between the n-th and (n+1)-th collisions will be decreasing as

For simplicity we assume that all proportionality factors are equal to one and that the reaction starts with two particles at time zero.

a.)Find the time of the 10th, 100th, 1000th and 10,000th collision.

b.)The reactor explodes if infinitely many particles are created in finite time. Explain in terms of the convergence or divergence of an infinite series why and when the explosion will occur.

c.)Use graphical and/or numerical methods to determine the number of particles in the reactor 0.05 time units before the explosion.

d.)With the help of a cooling system, all newly created energy can be dissipated. In this case the speed of the particles will not increase, but collisions will still be more and more frequent due to the increasing number of particles.

Will an explosion still occur? Explain your answer in terms of the convergence of divergence of an infinite series. If you conclude that there will be an explosion, give the first time by when the number of particles exceeds all bounds.

e.)How many particles will there be in the cooled system the time the uncooled system would explode?

Next: About this document Up: INFINITE SERIES Previous: 2.Convergence of infinite series

D. Vermes
Wed Jan 29 08:52:09 EST 1997