Power Series

Expansion of the Power Series 

In most of the textbooks, the expansion of the power series is not given but in this article, we will discuss everything in detail right from its origination to its expansion. Different variables are used in this sequence. Some of those terms are a,c,x, and n. 

Now, we will understand all these terms one by one. So, a stands for the terms of the sequence, x stands for the common factor if there is any, c stands for the centre of the series which is usually present in all series and is still taken to be zero and n stands for the power. 

Now the thing worth noticing is that all these variables are situation-based and change from one sequence to another sequence. Let us take an example of the sequence like 2,4,6,8…, so in this sequence a is the terms of the sequence which are the terms themselves, x is the number two as it is the only common factor present between all the numbers, c is zero as mentioned above and n is not clear as there is no case of power in the sequence. 

Mathematics involved in Power Series 

Some more examples

If we take the example of a sequence like 2,4,8,16,32…., we notice here that along with a being the terms themselves, x being two again, c being zero, n is also there because all the numbers in the sequence can be expressed as powers of two. The thing that makes this series sort of complicated and rare is the fact that all of the variables in any sequence are not always given and sometimes, we have a hard time finding them. 

Now some special sequences require the power part to be fractional, but one of the major rules of this sequence is that fractional powers are not allowed and so we have to make the powers negative which is generally not the case in many sequences. One more very important rule is that the sequence cannot involve any trigonometric variable like the sine function, cosine function, etc. 

The reason behind that is the fact that trigonometry involves very complicated topics like the concept of quadrants in which the sign and nature of the variables themselves become negative and positive simultaneously. Hence, to prevent this confusion, the series exempts from having trigonometric variables. 

Various types of operations on power series

Now on power series various operations like the addition, subtraction, multiplication, etc.  If we are to learn about the operation of the power series in-depth, then we will have to introduce two new variables f and g. Now, these variables are just two different cases, the variable f comes into play when we take the first term into consideration and the variable g comes into play when we consider the second term. 

Everything in the series is common except the first terms and that is the thing that we have to take common in separate brackets whenever we talk about combing terms independent of the fact that we are adding or subtracting or even multiplying. In this case, also, we have to take the value of c to be zero only. The variable c stands for the radius of convergence and as the power series does not end, we take the radius of convergence to be zero, if not mentioned otherwise. 

The concept of differentiation and integration 

Now we will read about how to perform the process of differentiation and integration on a special series. It was mentioned that we have to separate the functions f and g and put them in separate brackets. But if we are to perform integration and differentiation then we will have to separate them again and put one upon the other. 

After the integration or the differentiation, we have to add a constant k also which helps to include all the additional values that were supposed to be added on earlier. 

Miscellaneous series 

There is a lot of series that the students are taught early on like arithmetic progression, geometric progression, and harmonic progression. The arithmetic progression also consisted of terms like a,d, and n wherein a stands for the first term, d stands for the common difference between the terms and n stands for the number of terms. 

The harmonic progression is the reciprocal of the arithmetic progression, so all the operations can be applied on the same methods and the patterns themselves. In the case of geometric progression, we will find that there is a new variable introduced namely r which replaces d from the arithmetic and geometric progressions. The variable r stands for common ratio.

Conclusion 

Power series is one of the very rare series that exists. Even though it does not end, it involves loads of variables that make it very difficult to understand. All of the arithmetic operations also do not apply to this series.