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Poopsilon
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I can't seem to find the radius of convergence for this series, or even a suitable function that bounds it for certain values of z. The series is £ 1/(1+z^n) sorry for the pound sign I'm on an iPod touch. Thanks.
Poopsilon said:I can't seem to find the radius of convergence for this series, or even a suitable function that bounds it for certain values of z. The series is £ 1/(1+z^n) sorry for the pound sign I'm on an iPod touch. Thanks.
Poopsilon said:How would I go about proving that, I can't seem to get the ratio or root test to work. I could bound it by 1/z^n which does converge outside the circle of radius 1 but bounding it is only true when Re(z^n) >= -1/2, I think, and that seems to depend on z and n in a very complicated way.
carlodelmundo said:We are looking for the convergence of the series for differing values of z where possible values of z are from (-inf, +inf).
The Radius of Convergence for a series is a mathematical concept that determines the interval of values for which a power series will converge. It is denoted by the letter R and is typically a positive real number or infinity.
The Radius of Convergence is calculated using the ratio test, a method for determining the convergence or divergence of a series. The formula for the ratio test is:
R = limn→∞ |an+1/an| , where an is the nth term of the series.
The Radius of Convergence is important because it tells us the values for which a given power series will converge. It also helps us determine the interval of convergence, which is the set of all values for which the series is convergent. This information is useful in applications such as Taylor series, where having a convergent series is necessary for accurate approximations.
If the Radius of Convergence is infinite, it means that the power series converges for all values of the variable. This is known as a "convergent power series" and is often used in mathematical and scientific applications to approximate functions.
The Radius of Convergence is directly related to the convergence of a series. If the value of the series is within the interval of convergence, then the series will converge. If the value is outside the interval of convergence, then the series will diverge. Additionally, the Radius of Convergence can also tell us the type of convergence, such as absolute or conditional convergence, for a given series.