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Find the radius of convergence and interval of co…

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Problem 10 Easy Difficulty

Find the radius of convergence and interval of convergence of the series.
$$\sum_{n=0}^{\infty}(-1)^{\cdot} \frac{x^{2 n}}{(2 n) !}$$

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Calculus 2 / BC

Calculus

Chapter 8

Infinite Sequences and Series

Section 5

Power Series

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Video Transcript

in this question. Whereas to find the radius of convergence and the interval of convergence of the following series to do that we're going to use the ratio test by the ratio test. We first need to calculate the limit of absolute value of N plus one over N as N goes to infinity. In our case this is going to be limited absolute value of X two, n plus two Over 20 plus two factorial because that's what we get when we replaced and by N plus one multiplied by the reciprocal of AM which is going to be two N factorial, Wilbur next to the to the end. Now this is equal to the limit absolute value of X squared times two N factorial divided by 20 plus one factorial gives us one over two, n plus one Times two and Plus two. Now since eggs doesn't depend on N, we're going to take it outside the limit, we're going to get absolute value of X squared times the limit of one over two and plus one times 2 and plus two and this limit is equal to zero. So we're going to get absolute value of X squared times zero And this is going to be zero By the ratio test for the series to converge. We want this limit to be less than one and in our case this limit is less than one for all X. Mhm. And this means that our series converges for all X. Therefore the radius of convergence is equal to infinity and the interval of convergence is from negative infinity to infinity

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