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Ricky\# 45303161

Chronus\# 55456164

Tim\# 21631163

This page is entirely written for MATH V01 Assignment 7 question 3 (b).

The key to approach this problem is to realize that the series is absolutely convergent and the absolute convergence implies that the original series is also convergent. First, implement the ratio test on the series to find the interval of x so that the series absolutely converges. Then, carefully state that the original series also converges in the interval using the argument above. The calculation is fairly straightforward. So, it should not take too much time; however, patience is required to lead the conclusion.

This post entirely refers to MATH100 Assignment 3.

(a)

Here, I introduce a colony of a germs as an example. I suppose one can count the number of germs from the size of colony, and a  scientist measures the size of colony every hour. If the colony is supplied with enough space and food in a very large container, the colony will grow unboundedly and so, the number of germs are also growing unboundedly. While if the size of container is small, and only limited amount of food is provided, say every 5 hours, then the size of colony will reach a equilibrium size and so as the number of germs. This phenomenon corresponds to the divergence and convergence of a sequence where each term represents the number of germs (or the size of colony) and the indices corresponds to how many times the scientist measures the size of colony.

(b)

I let a spaceship embody the convergence and divergence of series. This spaceship gets acceleration every hour by emitting certain amount of hydrogen gas, and the amount gas changes every time. The acceleration of the spaceship is directly proportional to the amount, hence the velocity is proportional to the sum of the gas emitted.  If the spaceship goes infinitely fast as it emits gas infinite times, then it corresponds to the divergence of the series obtained from the sum of gas emitted. On the other hand, if the speed reaches a constant speed as the spaceship emits the gas infinite times, then it corresponds to the convergence of the series.

According to Einstein’s relativity,  the speed is limited by c, the speed of light; however, in this case, I don’t consider those effects.

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