Problem on convergent or divergent series

Question

Assignment:

Q1. Determine if the following series are convergent or divergent

Σ^∞_n=3 sin (nπ) / n
        
Q2. Find the values of x for which

L = lim_n→∞ |^an + 1/ ^an| < 1
 
and tell if the series converges or diverges when given the series

 Σ^∞_{n = 1} (x-5)ⁿ/n

Q3. Find power series for the following functions

x/1 +x⁴

Q4. Write down the first 4 Taylor polynomials around zero and plot f(x) a long with its approximation f(x)=e^2x+1

Q5. Use the integral test to determine if the series converges or diverges
∑^∞_k=1 1/2k - 1/(2k+1)   

Q6. Use the comparison test test to determine if the series converges or diverges

∑^∞_k=1 1/2k⁴+8

Q7. Evaluate the integrals.Use trig substitution.

∫√1-36x² dx

Provide complete and step by step solution for the question and show calculations and use formulas.