acdurbin953 said:Homework Statement
Find the interval of convergence of the power series ∑(x-2)n / 3n
Homework Equations
ρn = |an+1| / |an|
The Attempt at a Solution
I got that ρn = | (x-2) / 3 |. I set my ρn ≤ 1, since this is when the series would be convergent. Manipulating that expression, I got that the interval of convergence is -5 ≤ x ≤ 5. The answer in the back of the book is -1 ≤ x ≤ 5. I am confused where the -1 comes from.
Ray Vickson said:Be careful: the endpoints are generally not included, because they would correspond to a series of the form ##\sum 1^n## or ##\sum (-1)^n##. Are you sure the book did not say ##-1 < x < 5##?
The interval of convergence is the range of values for which a given mathematical series converges, or approaches a finite limit. It is usually expressed in terms of the variable or constants within the series.
To find the interval of convergence, you can use the ratio test or the root test, which are both methods of determining the convergence or divergence of a series. The interval of convergence is typically found by setting the ratio or root equal to the variable or constant within the series and solving for the values that satisfy the inequality.
The interval of convergence is important because it tells us for what values the series will converge and produce a meaningful result. It also helps us determine the behavior of the series at the endpoints of the interval, which can impact the overall convergence of the series.
Yes, the interval of convergence can change for a given series if the values of the variables or constants within the series are altered. This can also occur if a different method of finding the interval of convergence is used, such as the alternating series test.
If a series does not have a defined interval of convergence, it is said to have no interval or to diverge. This means that the series does not approach a finite limit and the values within the series do not have a consistent pattern or behavior.