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  • Combine power series by addition or subtraction.
  • Create a new power series by multiplication by a power of the variable or a constant, or by substitution.
  • Multiply two power series together.
  • Differentiate and integrate power series term-by-term.

In the preceding section on power series and functions we showed how to represent certain functions using power series. In this section we discuss how power series can be combined, differentiated, or integrated to create new power series. This capability is particularly useful for a couple of reasons. First, it allows us to find power series representations for certain elementary functions, by writing those functions in terms of functions with known power series. For example, given the power series representation for f ( x ) = 1 1 x , we can find a power series representation for f ( x ) = 1 ( 1 x ) 2 . Second, being able to create power series allows us to define new functions that cannot be written in terms of elementary functions. This capability is particularly useful for solving differential equations for which there is no solution in terms of elementary functions.

Combining power series

If we have two power series with the same interval of convergence, we can add or subtract the two series to create a new power series, also with the same interval of convergence. Similarly, we can multiply a power series by a power of x or evaluate a power series at x m for a positive integer m to create a new power series. Being able to do this allows us to find power series representations for certain functions by using power series representations of other functions. For example, since we know the power series representation for f ( x ) = 1 1 x , we can find power series representations for related functions, such as

y = 3 x 1 x 2 and y = 1 ( x 1 ) ( x 3 ) .

In [link] we state results regarding addition or subtraction of power series, composition of a power series, and multiplication of a power series by a power of the variable. For simplicity, we state the theorem for power series centered at x = 0 . Similar results hold for power series centered at x = a .

Combining power series

Suppose that the two power series n = 0 c n x n and n = 0 d n x n converge to the functions f and g , respectively, on a common interval I .

  1. The power series n = 0 ( c n x n ± d n x n ) converges to f ± g on I .
  2. For any integer m 0 and any real number b , the power series n = 0 b x m c n x n converges to b x m f ( x ) on I .
  3. For any integer m 0 and any real number b , the series n = 0 c n ( b x m ) n converges to f ( b x m ) for all x such that b x m is in I .

Proof

We prove i. in the case of the series n = 0 ( c n x n + d n x n ) . Suppose that n = 0 c n x n and n = 0 d n x n converge to the functions f and g , respectively, on the interval I . Let x be a point in I and let S N ( x ) and T N ( x ) denote the N th partial sums of the series n = 0 c n x n and n = 0 d n x n , respectively. Then the sequence { S N ( x ) } converges to f ( x ) and the sequence { T N ( x ) } converges to g ( x ) . Furthermore, the N th partial sum of n = 0 ( c n x n + d n x n ) is

n = 0 N ( c n x n + d n x n ) = n = 0 N c n x n + n = 0 N d n x n = S N ( x ) + T N ( x ) .

Because

lim N ( S N ( x ) + T N ( x ) ) = lim N S N ( x ) + lim N T N ( x ) = f ( x ) + g ( x ) ,

we conclude that the series n = 0 ( c n x n + d n x n ) converges to f ( x ) + g ( x ) .

Questions & Answers

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Adin
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Adin
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biomolecules are e building blocks of every organics and inorganic materials.
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research.net
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Introduction about quantum dots in nanotechnology
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nano basically means 10^(-9). nanometer is a unit to measure length.
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there is no specific books for beginners but there is book called principle of nanotechnology
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fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
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what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
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Mostly, they use nano carbon for electronics and for materials to be strengthened.
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carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
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Do you know which machine is used to that process?
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of graphene you mean?
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or in general
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in general
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Graphene has a hexagonal structure
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Source:  OpenStax, Calculus volume 2. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11965/1.2
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