Question 2 Use the Alternating Series Estimation Theorem to find the minimum number of terms of the infinite (-1)" series n3 we need to add to approximate the sum of the series with |error| < .008. n=1

I am having trouble understanding this series and its similar sequences. Considering the series in the image titled "1".  If a_n = ln(n + 1) − ln(n). Does the sequence {an}∞_n=1 converge or diverge? Additionally, considering the altered series in the image titled "2", how would I rewrite the expression by expanding the sum?  Lastly, how could I use the rewritten expression from before to show that this series diverges. Would this also be shown through the test for divergence?  Thank you, any help you can provide me with would be greatly appreciated. :)

Find the radius and interval of convergence of the given power series (-1)"n²(x + 2)" U. 3n

4. (a) How many terms of the given alternating series do we need to add in order to find the sum of the series correct to 5 decimal places? (That is, Jerror|< 0.000005.) Show your work! (b) Approximate the sum of the series correct to 5 decimal places. Show your work! 7 (-1)7+1. 2" (n!) n=1