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Unformatted text preview: that the series will converge will be a‐R < x < a+R. Here we denote R as the radius. Since we know that the ratio test and root test fail when the limit is equal to 1, we know that we need to test the endpoints using some other series test we know.
Step 1: Find the radius of convergence, and use this to construct an open interval that the series converges within.
Step 2: Test each endpoint independently to find out if it converges there, then use this to construct your final interval. Example 1 Continued xn
Determine the interval of convergence for the following: 2
n 1 n
We found the interval to be 1 for this series, so we can see that the center is at 0 (a=0) in this case, our interval so far is ‐1 < x < 1. This means we must test two series, when x = 1, and when x = ‐1: 1n
When x = 1 we get: 2
n 1 n
n 1 n We know this series converges by the p‐test, p = 2 > 1. This means we can include 1 in our interval of convergence. (1) n
When x = ‐1 we get: 2
n 1 n We know...
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