Math with Mrs. Sarah

(Finney, AP Calculus BC)

1. Write an expression for aₙ.
   a) ∑ aₙ = 1 + 1/3 + 1/9 + 1/27 + 1/81 + …
   b) ∑ aₙ = 1 − 1/2 + 1/3 − 1/4 + …
2. Determine whether the series converges by finding the limit of partial sums.
   a) 1 + 1.1 + 1.11 + 1.111 + …
   b) 3 + 0.5 + 0.05 + 0.005 + …
   
3. Let Sₙ = (n+1)/(2n+5) be the nth partial sum of the series Σaₙ. Write a rule for aₙ.
   
4. Tell whether the infinite series converges or diverges.
   a) 1 − 2 + 3 − 4 + 5 − … + (−1)ⁿ(n+1) + …
   b) ∑ (n=0 to ∞) (2/3)(5/4)ⁿ
   c) 3 − 0.3 + 0.03 − 0.003 + 0.0003 − … + 3(−0.1)ⁿ + …
   d) ∑ sinⁿ(π/4 + nπ)
   
5. Find the interval of convergence and the function of x represented by the geometric series.
   a) ∑ 3 · ((x−1)/2)ⁿ
   b) ∑ tanⁿx
   
6. Use differentiation to find a power series for f(x) = 2/(1−x)³
   
7. Find a power series that represents 1/x on (0,2). Hence find a power series for lnx centered at x=1.
   
8. Given Σ5xⁿ = q, solve for x in terms of q.
   
9. What conclusion can be reached by using the nth term test for divergence given the series
   a) ∑ (2n² + 13n + 15) / (n³ + 8n² + 16n)
   b) ∑ (−2)ⁿ

1. Write an expression for a(n).
   a) summation(a(n), n=0 to infinity) = 1 + 1/3 + 1/9 + 1/27 + 1/81 + …
   b) summation(a(n), n=1 to infinity) = 1 – 1/2 + 1/3 – 1/4 + …
2. Determine whether the series converges by finding the limit of partial sums.
   a) 1 + 1.1 + 1.11 + 1.111 + …
   b) 3 + 0.5 + 0.05 + 0.005 + …
3. Let S(n) = (n+1)/(2n+5) be the nth partial sum of the series summation(a(n), n=0 to infinity). Write a rule for a(n).
4. Tell whether the infinite series converges or diverges.
   a) 1 – 2 + 3 – 4 + 5 – … + (-1)^n * (n+1) + …
   b) summation((2/3)(5/4)^n, n=0 to infinity)
   c) 3 – 0.3 + 0.03 – 0.003 + 0.0003 – … + 3*(-0.1)^n + …
   d) summation(sin^n(pi/4 + n*pi), n=0 to infinity)
5. Find the interval of convergence and the function of x represented by the geometric series.
   a) summation(3((x-1)/2)^n, n=0 to infinity)
   b) summation(tan^n(x), n=0 to infinity)
6. Use differentiation to find a power series for f(x) = 2/(1-x)^3
7. Find a power series that represents 1/x on (0,2). Hence find a power series for ln(x) centered at x = 1.
8. Given summation from n=0 to infinity of (5x^n) = q, solve for x in terms of q.
9. What conclusion can be reached by using the nth term test for divergence given the series?
   a) summation((2n^2 + 13n + 15) / (n^3 + 8n^2 + 16n), n=0 to infinity)
   b) summation((-2)^n, n=0 to infinity)