The following integrals were solved by using integration techniques such as substitution, integration by parts and partial division.

This pictures were taken from latex, where we type the exercises.

We always think integral and an antiderivative are the same thing. However, I prefer to say that antiderivative is much more general than integral. Specifically, most of us try to use antiderivative to solve integral problems and just view them as the same thing. In fact, further studies imply that there are many situation that we can not apply antiderivative to solve such problems. For example:∫x4−3×2+6×6−5×4+5×2+4????????21. When apply antiderivative, we get the answer ????????????????????????(x2−3xx2−2)+C(by Wolfram Alpha). As the function is not continuous at √2, it is only an antiderivative but not a integral.

An antiderivative refers to a different function whose derivative is the original function. What we mean with that is that if we have a function F'(x)=f(x) in a given interval. For example F(x)=????6, then F’(x)=17×7=f(x). But notice that17????7+C, where C is a constant, can also be the antiderivative of ????6. So us the illustration shows below jellyfish have same function but slightly different sizes, which represents the constants that each family of antiderivative could have. https://www.google.ca/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwjJupHy7eLRAhVHKWMKHaIADdMQjRwIBw&url=https%3A%2F%2Fwww.pinterest.com%2Fcmariegatlin1%2Fjellyfish%2F&psig=AFQjCNEcSNC0GFsWCBwrYCRTas-cIkXemw&ust=1485624743362004

However, integral can either be the antiderivative of an indefinite integral or an actual number, which means the “anti-derivative” and the “indefinite integral” are the same thing, but the “definite integral” is the area under the curve which is bounded by l and r, which can be actual number.

We all learn about Riemann Sum. We can divide the domain into infinite vertical lines (making infinite subintervals) for Riemann integral in one dimension. We can use rectangular area formula to figure out the value.

So in some general explanation the main differences are:

Antiderivative talks about undoing differentiation, integrals assign number that tell us how big it is.

A definite integral has limits of integration which give us a number as an answer, while an antiderivative give us a function in terms of the independent variables.

It is worth mentioning that there is a Fundamental Theorem of calculus we can use at Riemann integral. It says if G(x) is an antiderivative of g(x). Then we can use G(x) to express the integral of g(x). The Fundamental Theorem of Calculus connect the integral with antiderivative.

But in other dimension( 3-dimension). We can not use this Theorem. We can not find antiderivative of function. At that time, there will be other Theorem we can use———Stokes Theorem and etc.

During the test it is very difficult to think clearly if you are so stress, so my main recommendation is to calm down, and read very well the problem.

For this question in particular it was necessary to understand very well, what I was reading. First understanding what kind of  data do I have it and then, what the questions were asking for. Then in order to figure it out which was the pattern or sequence that follows the carpet, it was necessary to put attention to how was the main trend of the graph. Then by applying the studied concepts, it was more easy to find out the answer. Specially if you think in which is easiest kind of series to work with, while doing sum.

Following the questions in the given order was also very helpful, because, by going from a specific pattern to a more general one was the key point in solving the rest of the questions.  For the second question, Just applying previous knowledge was OK. An finally for the third question the most important thing was to visualizate how the trend or sequence works  and think about ,what if it goes to the infinity?.

DNA 🙂

Hi everyone,

In order to complete the assignment from my Math course I have to talk about of series and sequences. I have been struggling with them for three weeks now, so What I  learned about  this topic is as follows.

• One of the first things that I had to understand was the fact that series and sequences are not the same. Pretty similar but not the same. So We might not treat  them as equals. However, they shared a relationship  between each other. In order to do this I am going to put some examples that have help me to understand this topic.

• First at all, We can say that a sequence is a bunch of numbers with the same characteristics or a special arrangement, while series are the same bunch of numbers but added each other. In order words, or in my words I would say that it is like in a classroom, we can say that the sequence is for example the list of the people that belong to a same class, they could be arranged by their last name, student number etc.  And the series is equal to the total number of students in that class.

Now the difficult part, How can I understand when  sequences  and series converge or diverge?. Well, Lets start by taking  each one by themselves.

1.  The convergence of a sequence is difficult to set, but not impossible. First we can think in other simple example for a sequence. Let say that a sequence is like having  cats in a room. For this case I have to possibilities.

• If I left the door close, any cat is going to escape from my room, they can touch the walls and the door, but They are never going to get out of the room. Which means that they are bounded to a given space. Similar to what happen with sequences, if the are bounded to an specific interval, that means that they converges, also they approaches and touch the edges or the limits of that space, but the are never going to be point out of that space.
• Another variation of this could be that the cats are scattered in my house and if I put some food on the room, they are going to approach more and more to the food, even knowing that I am going to locked the door. So what i mean with this, is that in a sequence, the sequences converge also, if they are approaching to one number.
• For example: an= 1/n= 1, 1\2, 1\3, 1\4…….1\n. We can say that it goes to the infinite. Therefore We can see that the number get smaller and smaller, but they never pass the edge of {1 to 0} being 1 the maximum number or my upper bound and 0 my lowest number or my lower bound. Furthermore, the values are approaching to 0, so I an say that this sequence converges to 0

• For Instance: an= 2n = 2,4,6,,8,10,12…… 2n . Where the values are getting bigger and bigger, or if I choose negative numbers, they get smaller and smaller, but they always go to the infinite, and in this case they are not approaching to any number.
• In the second case, If I forget to close a window or the door of that room, eventually the cats are going to get out the room, and they will be scattered in all the house, and even in all the neighbourhood. So I can say that this is a good example of what represent to have a divergent sequence. Mainly because, in a divergent sequences the number are out of edges or limits. What I mean with this, is that in divergent sequences even if I go to infinity I will never be able to lock up the numbers in the same space.
• Furthermore, in the case I never open the door, the cats are going to be always in the same room, but if I put some water in the middle of the room as  a curtain. The cats are going to be so afraid that they are never going to pass that curtain. Some cats are going to be in the right, and some others on the left, and they are not going to be together again. With this extreme case, I want to prove that if a sequence has two different limits, the whole sequence diverges.
• One example of this is: (-1)^n+1= 1,-1,1,-1,1,-1,1,-1…., and so on. In this particular example the sequence diverges, because the numbers have different limits.

1. 2.  The convergence of series is totally different from, what we have said before. It is quite more difficult, But I am going to try to explain this one, also with cats.

• For instance : the SUM of  1\2^n= 1/2 when n=1 is a convergent series, because if I sum the terms the are going to converge to 1/2.
• As I am experimenting with cats, I am going to divide my cats in two sections. In one group I am going to put just female cats and in the other I am going to put male cats. I am going to be a terrible person, so I am going to left then alone for two weeks, without food or water. 🙁 Eventually the cats are going to die, and the sum of my cats is going to decay. Even If I can save one of then I am not going to have the same amount as in the beginning. This is what happens with the sum of my series with certain characteristics, if I compare the sum of a serie with other  that I can say is bigger  it  means that converges to a number, just equal to the cats. Even If I put couples of cats, male and female, in some point they will have kittens, but if I forgot to feed them a week, the sum of them will converge to the number of  alive cats.
• In other case, I am going to put my cats all together, male and female. An this time I am going to be responsible and feed them. In some point my cats are going to have kittens, so the sum of the cats in my room is going to increase and increase. If I leave them forever, at the end I am going to have lot of kittens (imagining that this is possible). The same way series diverge, what I mean with this is that eventually the some of my terms is going to go to the infinity, and that can not be possible to count obtain a sum.
• For example: SUM of 9^n/8^n when n=1, is a divergent series, because the sum of its terms is going to be infinite.

Well in the last part there are some variations, but It is difficult to catch all them up, HOWEVER If you keep thinking about the cats, there are lot of cases  for experiment with them. Just In case That was just  a metaphor.

In conclusion I would say that, series and sequences seem so boring and difficult , but in real world they may contain the secret of the universe. 🙂

Hi World

My name is Daniela, I am 19 years old and I am from Ecuador. Ecuador is a little country in South America, but with a big heart and beautiful landscapes.

Nice day to start a blog.

DNA 🙂