Mark44 said:x = - 1 doesn't give the same series as x = 1. If x = -1, the series is
[tex]\sum_{n=1}^{\infty} \frac{(-1)^{2n-1}}{(n+1)\sqrt{n}}[/tex]
Since 2n - 1 is odd for all integers n, the numerator of the expression in the summation is always -1. If x = 1, the numerator is always +1.
Ratio Test =) said:No No Sorry!
there was a typo!
its 2n-2 not 2n-1 in the series =)
Sorry Again.
Ratio Test =) said:Thanks you.
you saved my life !
It is easy for me.
It converges by integral test.
Ratio Test =) said:Ohhh
My bad
How in Earth I did not notice this :@
The interval of convergence for a power series is the range of values for which the series converges. It can be found by using various convergence tests such as the ratio test or the root test.
The radius of convergence is calculated by finding the limit of the ratio of consecutive terms in the power series. This limit is known as the convergence ratio or the radius of convergence. The series will converge within this radius and diverge outside of it.
Yes, it is possible for a power series to have an infinite radius of convergence. This means that the series will converge for all values of the variable. This is usually the case when the series is a polynomial or a geometric series.
The endpoints of the interval of convergence can be determined by plugging in the endpoints of the interval into the original power series. If the series converges for both endpoints, then they are included in the interval of convergence. If the series diverges for one or both endpoints, then they are not included in the interval of convergence.
Yes, there are other methods such as the integral test and the comparison test that can also be used to determine the interval of convergence and the radius of a power series. However, these tests are not as commonly used as the ratio test or the root test.