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jack.o
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Is (3n+3)!=(3n)!+3!
? probably obvious but I'm not certain. Trying to work out a radius of convergence for a series.
? probably obvious but I'm not certain. Trying to work out a radius of convergence for a series.
jack.o said:Is (3n+3)!=(3n)!+3!
? probably obvious but I'm not certain. Trying to work out a radius of convergence for a series.
The radius of convergence for a series is a value r such that the series converges for all values of x within a distance of r from the center of the series, and diverges for all values of x outside of this distance.
The radius of convergence is typically found by applying the ratio test, in which the limit of the absolute value of the terms of the series is taken as n approaches infinity. If this limit is less than 1, the series converges, and the radius of convergence can be calculated as the reciprocal of the limit. If the limit is greater than 1 or infinite, the series diverges and the radius of convergence is 0.
The radius of convergence is an important value for determining the behavior of a series. If the value of x falls within the radius of convergence, the series will converge and the terms will approach a finite value. If x lies outside of the radius of convergence, the series will diverge and the terms will approach infinity. The radius of convergence also helps to determine the interval of convergence for a series.
The series 3n+3 and 3n+3! have the same radius of convergence, which is equal to 1. This means that both series will converge for all values of x within a distance of 1 from the center of the series and diverge for all values of x outside of this distance. However, the intervals of convergence may differ for these two series.
If the radius of convergence for a series is equal to 0, then the series will converge only at the center point and will diverge for all other values of x. This is because the radius of convergence represents the distance from the center of the series within which the series will converge. If the radius is 0, then the series will only converge at the center point and diverge everywhere else.