Find the interval of convergence of the power series. a.) (...

Transcript

00:01 In this question, we are asked to find the interval of convergence of the given power series.

00:06 And to do that, we will use the ratio test.

00:11 By the ratio test, we need to calculate the limit of the absolute value of an plus 1 over an, where an plus an is the general term of the series.

00:24 So this is an.

00:24 And to get an plus 1, we simply need to replace n by n plus 1 in the formula for the general term of the series.

00:25 So this is an.

00:27 And to get an plus 1, we simply need to replace n by n plus 1 in the formula for the general.

00:36 So this is what we are going to get.

00:59 And we need to calculate the limit of the absolute value as n goes to infinity.

01:04 Let's simplify the expression.

01:08 We can rewrite that fraction as x minus 3 to the n plus 2 over n plus 2 times 4 to the n plus second power multiplied by the reciprocal of the denominator.

01:32 Then we can do some cancellations.

01:35 We can cancel 4 to the n plus 1 power.

01:38 And we can cancel x minus 3 to the n plus first and we are going to get the limit of the absolute value of x minus 3 multiplied by n plus 1 over n plus 2 and as n goes to infinity n plus 1 over n plus 2 goes to 1 and in the limit we are going to get the absolute value of x minus 3 and by the ratio test, the series converges if that limit is less less than one.

02:19 So the series converges for the absolute value of x minus 3 less than 1.

02:23 However, the ratio test doesn't tell us what happens when the limit equals to 1.

02:29 So we have to check that case separately.

02:33 We have to check what happens when x minus 3 equals to negative 1 and x minus 3 equals to 1.

02:43 We need to test the series directly in these two cases...

Question

Find the interval of convergence of the power series. a.) ( sum_{n=0}^{infty} frac{(x-3)^{n+1}}{(n+1) 4^{n+1}} ) b.) ( sum_{n=0}^{infty} frac{(-1)^{n} x^{2 n}}{n !} )

Question

Find the interval of convergence of the power series. a.) ( sum_{n=0}^{infty} frac{(x-3)^{n+1}}{(n+1) 4^{n+1}} ) b.) ( sum_{n=0}^{infty} frac{(-1)^{n} x^{2 n}}{n !} )

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