Problem 41 Medium Difficulty

Suppose the series $ \sum c_n x^n $ has radius of convergence 2 and the series $ \sum d_n x^n $ has radius of convergence 3. What is the radius of convergence of the series $ \sum (c_n + d_n) x^n? $

Answer

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Video Transcript

since this sum here has a larger radius of convergence and this sum, we know that these dian terms we're going to be approaching zero significantly faster than these see in terms. So much so that as n goes to infinity, Deanne overseeing is going to go to zero. Now we can use the fact that our radius of convergence is the limit as n goes to infinity of absolute value of C N plus stian over C n plus one plus D in class one. And now we can divide both sides both the top and the bottom by CNN to get one plus D and overseeing over C N plus one oversea end plus the end plus one overseeing as we mentioned, since these dian terms go to zero much faster than these see in terms, this is going to go to zero. This is going to go to zero and then we'LL just have one divided by C n plus one over seeing plus one. So division is the same thing as multiplying by the reciprocal and so we get here. Okay, but this should be the radius of convergence for this sum here, which we set is too. Okay. So in fact, whenever you have a sum, some power Siri's and some other power Siri's and you're adding them up in this way, that radius of convergence for the sum is just gonna be the smaller of the two unless they happen to have the same radius of convergence, In which case, there's not quite as much to say.