Now let us take a look at a case where we find the interval of convergence using the root test. r = 0 implies the power series is convergent for all x values, and r = ∞ \infty ∞ implies the power series is divergent always. Notice again that you will encounter cases where you will end up either getting r = 0 < 1 or r = ∞ \infty ∞. If it is convergent, then include it in the interval. Don't forget to check the endpoints to see if the series is convergent or divergent. How do we find the interval of convergence using the root test? Well again, we just use the root test formula and set r<1and try to get the inequality |x-a| interval of convergence will be. Hence the radius of convergence is infinity, and the interval of convergence is - ∞ \infty ∞ < x < ∞ \infty ∞ (because it converges everywhere). This test requires you to calculate the value of R using the formula below. Now using the root test, we will have that: Equation 3. This means that for all x values, the power series converges. Ratio test is one of the tests used to determine the convergence or divergence of infinite series. However notice that r = 0 < 1 for all x values. You may see that this is not in the form |x-a| < R. Equation 4: Ratio test Interval of Convergence pt.
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