assignment 5

Assignment 5 Question 3

Chenming Hu 1797365 / Qiwei Liang 33196163

Alphonse Wang 47168166 / Yuqi Liu 44616167

Motivation of Integration

Before 370 BC, ancient Creek Astronomer Eudoxus had an idea to find the specific areas and volumes by breaking them up into small pieces of area for which the area and volume was known. This method was further developed and employed by Archimedes in 3^rd century BC and used to calculate approximation to the area of a circle. And over last thousand years, the way of using integral has been promoted by lots of scientist. During these thousand years’ promotion, a famous scientist Isaac Newton used a small vertical bar above a variable to indicate integration. However, the box notation was difficult for printers to reproduce, so these notations were not widely adopted. Finally, in 1675 Gottfried Leibniz create the modern notation for the indefinite integral.

Today ‘Integration’ as a term is widely applicated as different meaning in different disciplines such as Sociology, economy and Engineering. Etc. In mathematics, ‘integration’ usually is represented as ‘integral’ which utilize a function to assign number to represent area, volume and other concepts by breaking them up into an infinite number of division. For example, using integrals to find out the different value. For some functions, some parts are positive and above the x-axis, and some parts are negative which are below the x-axis. When we try to figure out the average of these values, we can use the graph, and add the positive areas and negative areas together and find out the difference between two values. Using graph can give a visual feeling directly and help researchers.

Moreover, people use ‘integral’ to determine the area under a curve and do the data analyzing in many field, typically, in physics. In physics, people always use a function to represent the relationship between speed and time. From this kind of function, we are not only looking the changing of the speed, but also the area under the curve, which can represent the displacement of the object in this time period. When we analyze the function, we can find out the moving state and position state of this object. What is more, when the function is about the acceleration and time. The area under the curve is about the speed, and by analyzing the acceleration and speed of the object, we can figure out the position. Therefore, the data can help us find more data. Both of these two examples relate the integral and the real life together, and it shows the integrals exist in many fields.

All in all, integration has already connected with our life and acadamic study. So hopefully, right now you have the motivation to learn ‘INTEGRATION’.

The question 4 includes 3 small questions. Let’s solve it step by step. First of all, question a asks to use the Ratio Test, so thies question must be connected with Ratio Test. Accoding to Ratio Test, we have find the rate of change in the series of sequence. If the rate of change is smaller that 1, definitely the sequence will become smaller and the series will be converges. For solving question b, we can use comparison test, whcih cound be used to distinguish whether the function converges or not by using the result we get from question a.  Compare with question b, the only difference is there is negative possibility in question c, which means the Alternating Series Test can be used in this question.

a) What distinguishes convergent sequence from divergent sequence?

If n is infinite and the sequence is convergent, the sequence is close to a real number. For example, the sequence {1/n}. Since the n is the number which is approaching to infinite, 1/n is sufficiently close to 0.

IF n is infinite and the sequence is divergent, the sequence does not approaching to any real number or approaching to -&+ infinity. It is not percise. For example, the sequence {-1^n}. If n is even number, the sequence will be positive 1. If n is odd number, the sequence will be negetive 1. And the sequence {3^n} is divrgent and it is approaching to the positive infinity.

b) What distinguishes convergent series from divergent series.

Series means the sum of the sequences. So convergent series means the sum of the sequences has limit. Namely it is sufficiently close to a number. On the contrary, divergent series meas the sum of the sequences is apporaching to the infinity.

Moreover, we can transfer the sequnence into the form  a*r^n. If the  |r|<1, the series is convergent. Since when n goes bigger, the sequence becomes smaller. And the sum of the series has limit. On the other hand, if the |r|is bigger than 1, the series is divergent. Same reason, when n goes bigger, the sequence becomes bigger, the sum of the series keeps increasing to the infinity.

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