Weekly Assignment 8

Instructions: Don' t leave these to the last minute -- they aren't a cake walk! 
Feel free to reach out during office hours for hints and help!:)

1. Comparison test

For each of the given series, determine if it converges or not. Give reasons for your answer. If it converges, approximate it with error less than 0.0001.

(Underoverscript[Σ, (k=0), ∞])((3^k)/((k^4)+(4^k)))

(Underoverscript[Σ, (k=0), ∞])((5(k^2)+k+1)/(2(k^3)+(k^2)+10))

(Underoverscript[Σ, (k=0), ∞])((Sqrt[9(k^2)-k])/(6(k^4)+7))

2. Alternating series

Consider the series (4/1)-(4/3)+(4/5)-(4/7)+(4/9)-…. This is a convergent series by the alternating series test. Using an appropriate error estimate determine how many terms are needed of this series to approximate the infinite sum with error less than 0.0001.

Another series that converges to the exact same value as the series above  is (Underoverscript[Σ, (k=0), ∞])(((((-1))^k)2)/((2k+1)(3^(k-1/2)))). Using an appropriate error estimate determine how many terms are needed of this series to approximate the infinite sum with error less than 0.0001.

Choose either of the above two series and approximate it with error less than 0.0001. What famous number do these two series series converge to?