MHB Calculus 2 (Power Series) when the limit is zero by root test

[yeny]

Hi guys!

Here's a problem i was working on. I solved it by root test and got absolute value of x on the outside of the limit and the limit equaled zero. Is it wrong to multiply the outside absolute value by the zero I got from the limit? or is that okay?

In general, when we are solving power series problems, is it okay to think of R equals infinity when the limit is zero? is that always the case? the interval of convergence is (-inf, +inf)

what are the steps that YOU would take to solve such a problem?

hope this makes sense. THANK YOU !

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[Opalg]

Hi guys!

Here's a problem i was working on. I solved it by root test and got absolute value of x on the outside of the limit and the limit equaled zero. Is it wrong to multiply the outside absolute value by the zero I got from the limit? or is that okay?

In general, when we are solving power series problems, is it okay to think of R equals infinity when the limit is zero? is that always the case? the interval of convergence is (-inf, +inf)

what are the steps that YOU would take to solve such a problem?

hope this makes sense. THANK YOU !

The root test only works for series of positive terms. So you should start by taking the absolute value of $|x|$. When you find that the series converges you can then use the fact that absolute convergence implies convergence to deduce the result for negative $x$.

Apart from that, your answer is correct. The limit of the $k$th root is zero, so you can conclude that the series converges (absolutely) for all $x$. That is expressed informally by saying that the radius of convergence is infinite.