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jaykobe76
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Homework Statement
how to prove that radius of convergence of a sum of two series is greater or equal to the minimum of their individual radii
i don't know how to begin, can someone give me some ideas?
kai_sikorski said:Hi Jay,
Welcome to the Forums!
I wouldn't try to do this by working with the coefficients of the series directly. Instead fix some x within the radius of convergence of both sequences. Define fn(x) to be the quantity that x gets mapped to by the first n terms of the first power series, and gn(x) the same thing except for the second series. What do you know about the sequences
f1(x), f2(x), f3(x), ... and
g1(x), g2(x), g3(x), ... ?
Using that what can you say about the sequence (g + f)n(x)?
kai_sikorski said:Oh if you're asking why it might be greater, then it's simple because your proof excluded the possibility that
R1 + 2 < min(R1, R2)
So it must be true that
R1 + 2 ≥ min(R1, R2)
That means that it might be greater maybe, or it might be equal. And in fact you could find examples for both.
sorry, i think i get it . because from my proof that |x|<min{R1,R2} then it means that the new radius K cannot smaller than {R1,R2}so K>=min{R1,R2} is this the idea?kai_sikorski said:Oh if you're asking why it might be greater, then it's simple because your proof excluded the possibility that
R1 + 2 < min(R1, R2)
So it must be true that
R1 + 2 ≥ min(R1, R2)
That means that it might be greater maybe, or it might be equal. And in fact you could find examples for both.
jaykobe76 said:sorry, i think i get it . because from my proof that |x|<min{R1,R2} then it means that the new radius K cannot smaller than {R1,R2}so K>=min{R1,R2} is this the idea?
jaykobe76 said:then by definition of convergent we have |fn(x)-T|<esillope/2 for |x|<R1
kai_sikorski said:I guess here technically you should say that there exists an N, such that this is true for all fn(x) with n>N.
kai_sikorski said:I guess here technically you should say that there exists an N, such that this is true for all fn(x) with n>N.
The radius of convergence problem is a mathematical concept that involves determining the interval or region of values for which a given power series converges. It is an important topic in calculus and analysis, and is often encountered when working with infinite series.
The radius of convergence is typically calculated using the ratio test, which involves taking the limit of the absolute value of the ratio of successive terms in the series. The result of this limit will determine the radius of convergence, which can be either a finite value or infinity.
If the radius of convergence is infinite, it means that the series converges for all values of the variable. This is also known as a convergent power series. In other words, the series will approach a finite value as the variable approaches infinity.
No, the radius of convergence cannot be negative. It is always a non-negative value, including zero and infinity. A negative value would not make sense in the context of determining the convergence of a series.
The radius of convergence is used in various fields of science and engineering, such as physics, chemistry, and economics, to model and analyze real-world phenomena. It is also used in computer science to approximate functions and in statistics to estimate values. In general, it is a useful tool for understanding and predicting the behavior of functions and series.