Determine the radius of convergence of the power series below. Then test the endpoints, if necessary, to determine the interval of convergence. express in interval notation

Question

the sum as k=0 goes to infinity of, (x+4)^k/(k+1)•7^k

Answer 1

I tried a video submission but for some reason it did not upload. It would be more complicated to write it out but I will do my best to make it clear. Feel free to message me and set up a quick session to go through it together live if you have any questions!

First, we are going to use the Ratio Test which recall is lim k →∞ |a_k+1/a_k|

Instead of having fractions over fractions I'm simply going to do a keep change flip and multiply by the reciprocal of a_k.

so we have lim k →∞ |(x+4)^k+1/(k+5)^7k+7×(k+4)^7k/(x+4)^k|

we can split the powers on the (x+4) and get lim k →∞ |(x+4)^k(x+4)/(k+5)^7k+7×(k+4)^7k/(x+4)^k|

Then (x+4)^k cancels leaving us with lim k →∞ |(x+4)/(k+5)^7k+7×(k+4)^7k|

Since our limit deals with k going to infinity, we can pull out the x+4 but we must make sure we remember to put absolute values.

|x+4| lim k →∞ |(k+4)^7k/(k+5)^7k+7|

We know that lim k →∞ |(k+4)^7k/(k+5)^7k+7|=0 (by calculation, observation that the bottom is larger, or graphing utility)

Finally when using Ratio Test, we have to make sure that what we get is less than one by convergence criteria.

So |x+4|×0<1

All values of x satisfy this inequality since anything times zero is zero which will always be less than 1.

Thus our radius of convergence is infinite so R=∞

We have no endpoints to test so we are done!